Reliability Assessment of Damaged RC Moment-ResistingFrame against Progressive Collapse under Static

Loading ConditionsZhiwei Huang1; Bing Li2; and Piyali Sengupta3

Abstract: Prevention of structures against progressive collapse has become a concern of increasing significance and the alternative load pathmethod (ALPM) is the extensively acknowledged approach in this area of research.With the basis of ALPM and by amalgamation of theMonteCarlo simulation method and an iterative algorithm, a new approach is developed in this paper to assess the reliability of ductile RC framestructures when subjected to one column failure. This paper focuses on the construction and solution of the performance function of thedamaged structure under static service loadings. With this intention, three types of critical zones are identified and an optimum model is pre-sented to determine the possible failuremodes for each zone. By employing the virtual work principle and critical collapsemechanism criterion,the performance function is established in terms of minimum internal virtual work and external virtual work done by service loadings. Threeequations containing the interaction between the axial forces and bending strength of structural members are evolved to calculate this function,and by solving these equations from the top floor to the floor of the failed column the performance of the damaged structure can be appraised.DOI: 10.1061/(ASCE)EM.1943-7889.0000455. © 2013 American Society of Civil Engineers.

CE Database subject headings: Progressive collapse; Reinforced concrete; Frames; Static loads.

Author keywords: Reliability assessment; Progressive collapse; Performance function.

Introduction

Recent events of progressive collapse of structures reveal thestructural vulnerabilities under abnormal loading, which are notexplicitly considered in design. To prevent structural failure in suchsituations, various methods such as the event control method,specific load resistance method [Biggs 1964; Newmark 1972;U.S. Department of the Army (USDA) 1986, 1990; May and Smith1995] alternative load path method (ALPM) (Pretlove et al. 1991;Kaewkulchai 2003), and indirect design method (Marjanishvili2004; Kaewkulchai and Williamson 2006) have been adopted.Because ALPM is event independent and comprehensive, it hasbeen implemented in many design guidelines [ASCE 1995, 2002;European Committee for Standardization (CEN) 1996; GeneralServices Administration (GSA) 2000; Department of Defense (DoD)2001]. The progressive collapse resistance of RC structures was il-lustrated bySasani et al. (2011) by incorporatingfiber plastic hinges inthe analytical model. Ellingwood and Dusenberry (2005) criticallyaddressed various available approaches in their research for furtherenhancement of design practices. However, the majority of theprevious studies addressed this issue in a deterministic manner where

all parameters were assumed to be well defined and deterministic. Inreality, the random variation of some parameters including theapplied load, strength, and stiffness of all construction materials andthe structural geometries are inevitable and have a significant in-fluence on the structural safety. Therefore, structural reliabilityassessment based on probabilistic theory should be themost sensibleway to evaluate structural safety. For this purpose, a performancefunction for the damaged structure has been set up that can properlyreflect the structural failure event and calculate its occurrencepossibility. This paper focuses on the construction and solution ofthe performance function for damaged ductile RC frame structuresunder static service loadings. A new practical approach combinedwith the Monte Carlo simulation (MCS) method is presented for thereliability assessment of a damaged structure. A simple exampleof a 4-story RC frame structure subjected to two different cases ofcolumn failure is addressed to show its application.

Structural Modeling

The lumped plasticity model as shown in Fig. 1 has attested its goodperformance in structural progressive collapse analysis for ductileframe structures (Kaewkulchai and Williamson 2006). In Fig. 1(c),Mu indicates the end spring strength to simulate the nonlinear be-havior of plastic hinges during structural response. After failure andremoval of a column, a chain of load redistribution occurs within theremaining structure, inducing additional damage to other elements.If the structure is not sufficiently robust, it will lose its balanceentirely and result in progressive collapse. To measure the occur-rence possibility of structural collapse in such a situation, a criterionthat can suitably reflect the global system failure needs to be de-lineated. Under static loading conditions, the singularity of thestructural stiffness matrix is considered to satisfy this requirement.

1Senior Engineer, J P Kenny London, J P Kenny Ltd., Compass Point,79-87 Kingston Rd., Staines TW18 1DT, U.K.

2Associate Professor, School of Civil and Environmental Engineering,Nanyang Technological Univ., 50 Nanyang Ave., Singapore 639798(corresponding author). E-mail: cbli@ntu.edu.sg

3Ph.D. Candidate, School of Civil and Environmental Engineering,Nanyang Technological Univ., 50 Nanyang Ave., Singapore 639798.

Note. This manuscript was submitted on January 22, 2010; approved onApril 18, 2012; published online onApril 20, 2012. Discussion period openuntil June 1, 2013; separate discussions must be submitted for individualpapers. This paper is part of the Journal of Engineering Mechanics,Vol. 139, No. 1, January 1, 2013. ©ASCE, ISSN 0733-9399/2013/1-1e17/$25.00.

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For a ductile frame simulatedwith the lumped plasticity model, thesingularity of the structural stiffness matrix indicates the formation ofa ductile collapse mechanism in the damaged structure with sufficientplastic hinges occurring in various structural elements. Depending onwhether the initially failed column is located on an internal axis or onan external axis, there are generally two types of structural progressivecollapse mechanisms (schematically plotted in Fig. 2). However, inthe second case if the damaged frame structure is extended withthe strength of the elements and service loadings within the extendedpart being zero, this structural progressive collapse can be treatedas a special case of the first type. Therefore, the performance functiondeveloped in the following sections is based on the initial failure ofan internal column and is applicable for both cases. For convenienceof subsequent discussions, the following definitions are given:1. There are three types of critical zones, Types I, II, and III

(indicated by superscripts I, II, and III in Fig. 3), in which theplastic hinges after formation possibly contribute to the struc-tural collapsemechanism. The total distance between the TypeI and III critical zones is called the damaged span, with thespans between the Type I and II critical zones and Type II andIII critical zones being called left damaged span l and rightdamaged span r, respectively.

2. The structural members in a critical zone are designated asBeam 1 (b1), Beam 2 (b2), Column 1 (c1), and Column 2 (c2),as shown in Fig. 3. Columns directly connected to criticalzones are called directly connected columns, whereas allothers are defined as indirectly connected columns.

3. In each critical zone, springs are located at both ends ofthe structural elements with their initial strength similar tothe related members’ strength. This may induce error in thestructural reliability assessment because real plastic hinges(RPHs) on structural beams may not form at the end sections.To distinguish between a (RPH) and a plastic hinge representedby the yielded end spring, the latter is referred to as the ESPH.

4. The strength of the real structural members and the end springsare denoted by Mur and Mu, respectively. If the bottom fiber(for a beam) or the left fiber (for a column) of a structuralmember is in tension, the related strength is considered aspositive strength andis indicated byM1ur orM1u. On the otherhand, M2ur and M2u are used to represent negative strength.Plastic hinges corresponding toM1ur ,M2ur ,M1u, andM2u aretermed as positive RPH, negative RPH, positive ESPH, andnegative ESPH, respectively.

The global failure of a damaged ductile frame is characterized bya ductile collapse mechanism where sufficient ESPHs occur withinthe critical zones. Therefore, it is reasonable to assume that thevirtual deformation corresponding to a given displacement dD fo-cuses on the rotations of the ESPHs. Considering the occurrencepossibility of structural failure event Pf as a measurement, thestructural reliability can be evaluated based on the virtual workprinciple as follows:

Pf ¼ Pðz, 0Þ ð1Þ

where

z ¼ dWint 2 dWext ð2Þ

Eq. (2) is the structural performance function, where dWint anddWext 5 internal virtual work done by the ESPHs and the externalvirtual work done by the service loadings. For z, 0, the structuralfailure event occurs, otherwise the structure remains safe. In ad-dition, z5 0 indicates that the structure reaches its ultimate state.

External Virtual Work

Virtual work dWext;i done by service loadings qiðxÞ in the ith storycan be obtained by

dWext;i ¼ðl0

qiðxÞ � dDðxÞ dx ð3Þ

where l5 length of the damaged span and dDðxÞ5 virtual dis-placement of qiðxÞ along the damaged span, expressed as

dDðxÞ ¼ dD � x=ll ð0 # x # llÞ ð4Þ

dDðxÞ ¼ dD ��12

x2 lllr

�ðll # x # lÞ ð5Þ

If qiðxÞ is uniformly distributed, substituting dDðxÞ into Eq. (3)produces

dWext;i ¼ 12ql;i � dD � ll þ 1

2qr;i � dD � lr ð6Þ

Fig. 1. Lumped plasticity model: (a) structural system; (b) beam-column element; (c) resistant function of end spring

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where ql;i and qr;i 5 uniformly applied service loadings on the leftand right damaged spans, respectively, and ll and lr 5 respectivespan lengths. The external virtual work dWext should be the sum ofdWext;i from the story of the failed column ðL1Þ to the roof story ðL2Þ.Thus

dWext ¼ PL2i2L1

dWext;i ð7Þ

Internal Virtual Work

In general, the number of possible collapse mechanisms is largedepending on the various locations of the ESPHs, and the structurewill be safe only when all possible collapse mechanisms are capableof providing adequate strength to keep the structure stable. Thus, todeal with this complicated problem, the critical collapse mechanismcriterion is espoused to identify theweakest collapsemechanism andcalculate the internal virtual work.

Critical Collapse Mechanism Criterion

The critical collapse mechanism criterion states that the damagedframe structure will fail if its weakest collapse mechanism, which

produces the minimum value of the performance function with agiven virtual dD, cannot pledge the safety of the structure. Math-ematically, this can be expressed as

z ¼ minðdWint 2 dWextÞ ¼ minðdWintÞ2 dWext ð8Þ

with

z, 00Ef and Ef0z, 0 ð9Þ

where Ef 5 structural failure event, with minðdWintÞ implying theinternal virtual work done by the ESPHs of the structurally weakestcollapse mechanism. Thus, Eq. (8) and (9) portray the sufficient andnecessary condition for the occurrence of the structure failure event.

Minimum Internal Virtual Work

The internal virtual work dWint is determined by rotations of theESPHs within the critical zones. Therefore

dWint ¼ Pcj¼1

dWint;j ð10Þ

Fig. 2. Collapse mechanisms of the damaged frame: (a) collapse mechanism for the damaged frame with an internal column initially failed;(b) collapse mechanism of the extended damaged frame with an external column initially failed

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where dWint;j 5 internal virtual work done by the ESPHs within

the jth critical zone and c5 number of critical zones. Minimizing

dWint produces

minðdWintÞ ¼ Pcj¼1

�min�dWint;j

�� ð11Þ

Possible Failure Modes of Critical Zones

Deformation of Critical Zones

The internal virtual work produced by a critical zone (dWint;j) is de-termined by the virtual rotations of the related ESPHs. However, thelocations of the ESPHs during the structural progressive collapse areunknown at this stage, and to cover all possible cases in deduction ofthe formula for minðdWint;jÞ the ESPHs are preliminarily assumed toform at the end springs of a critical zone. An arbitrary rotation a isapplied to determine the virtual rotations of the ESPHs, and therelationships between the virtual rotations of the ESPHs and dD=ll,dD=lr, and a are written in Eq. (12)e(14). For the Type I critical zone

gIc1 ¼ gIc2 ¼ jaj; gIb1 ¼ jaj; and gIb2 ¼ ja þ dD=llj ð12Þ

For the Type II critical zone

gIIc2 ¼ gIIc2 ¼ jaj; gIIb1 ¼ ja þ dD=llj; and

gIIb2 ¼ ja2 dD=lrjð13Þ

For the Type III critical zone

gIIIc1 ¼ gIIIc2 ¼ jaj; gIIIb1 ¼ ja2 dD=lrj; and gIIIb2 ¼ jaj ð14Þ

where gIc1, g

Ic2, g

Ib1, and gI

b2 5 virtual rotations of the ESPHs onColumn 1, Column 2, Beam 1, and Beam 2, respectively, for acertain critical zone type (here, Type I).

Optimum Model for the Minimum Internal Virtual Work

The internal virtual work by each type of critical zone is the sum ofthe products of the virtual rotations and the strength of the corre-sponding end springs.

dW Iint ¼ MI

u;b1jaj þ MIu;b2ja þ dD=llj þ

�MI

u;c1 þ MIu;c2

jajð15Þ

dW IIint ¼ MII

u;b1ja þ dD=llj þ MIIu;b2ja2 dD=lrj

þ�MII

u;c1 þ MIIu;c2

jaj ð16Þ

dW IIIint ¼ MIII

u;b1ja2 dD=lrj þ MIIIu;b2jaj þ

�MIII

u;c1 þ MIIIu;c2

jajð17Þ

where MIeIIIu;b1 , MIeIII

u;b2 , MIeIIIu;c1 , and MIeIII

u;c2 5 strength of the endsprings on Beam 1, Beam 2, Column 1, and Column 2 in varioustypes of critical zones. To obtain the minimum internal virtual workby each critical zone, an unconstrained nonlinear optimum modelis formulated as

require a

minimize dWint;j ¼ f ðaÞ ð18Þ

Fig. 3.Deformation of various types of critical zones: (a) details of a critical zone; (b) Type I critical zone; (c) Type II critical zone; (d) Type III critical zone

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where dWint;j 5 f ðaÞ5 objective function of a; the solutions of thisoptimum model are detailed in Huang (2008) and expressed byEq. (19)e(21). For the Type I critical zones

min�dW I

int

� ¼ minh�

MI2u;b1 þ MI

þu;c1 þ MI2u;c2

dD=ll;

MI2u;b2dD=ll

ið19Þ

For the Type II critical zones

min�dW II

int

� ¼ minh�

MIIþ u;b2 þ MII

þ u;c1 þ MII2u;c2

dD=ll

þ MIIþ u;b2dD=lr;M

IIþ u;b1dD=ll

þ MIIþ u;b2dD=lr;M

IIþ u;b1dD=ll

þ�MII

þ u;b1 þ MII2 u;c1 þ MII

þ u;c2

dD=lr

ið20Þ

For the Type III critical zones

min�dW III

int

¼ min

hMIII

2 u;b1dD=lr;�MIII

2 u;b2 þ MIII2 u;c1

þ MIIIþ u;c2

dD=lr

i(21)

The possible failuremodes for the Type I, II, and III critical zones areplotted in Fig. 4e6.

Performance Function

Based on the previous solution, the minimum internal virtual work ofa damaged frame structure can be obtained by substituting Eq. (19)e(21) into Eq. (11), and the performance function can be rewritten as

z ¼ minðdWintÞ2 dWext

¼ PL2i¼L1

min�dW I

int;i

þ PL2

i¼L1min�dW II

int;i

þ PL2

i¼L1min�dW III

int;i

2PL2

i¼ L1dWext;i

¼ PL2i¼L1

minh�

MI;i2 u;b1 þ MI;i

þ u;c1 þ MI;i2 u;c2

dD=ll;M

I;i2 u;b2dD=ll

iþ PL2

i¼L1minh�

MII;iþ u;b2 þ MII;i

þ u;c1 þ MII;i2 u;c2

dD=ll

þ MII;iþ u;b2dD=lr;M

II;iþ u;b1dD=ll þ MII;i

þ u;b2dD=lr;MII;iþ u;b1dD=ll þ

�MII;i

þ u;b1 þ MII;i2 u;c1 þ MII;i

þ u;c2

dD=lr

iþ PL2

i¼L1minhMIII;i

2 u;b1dD=lr;�MIII;i

2 u;b2 þ MIII;i2 u;c1 þ MIII;i

þ u;c2

dD=lr

i2PL2

i¼L1

12ql;i � dD � ll2

PL2i¼L1

12qr;i � dD � lr ð22Þ

where i5 ith story from L1 (the story of the failed column) to L2(the roof story).

The strength of the end springs is a function of the member axialforce that will experience various degrees of variation after the initialfailure of the column. Because the solution of Eq. (22) is notstraightforward it is, hence, a new structural system; i.e., the con-struction system is created by applying an additional virtual con-centrated force vector Pv 5 fpv;ig to the damaged frame at the nodesabove the failed column

dWtext ¼ dWext þ dWpv

ext ð23ÞFig. 4. Possible failure modes of the Type I critical zone: (a) firstpossible failure; (b) second possible failure

Fig. 5. Possible failure modes of the Type II critical zone: (a) first possible failure mode; (b) second possible failure mode; (c) third possiblefailure mode

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zC ¼ minðdWintÞ22dWtext ¼

PL2i¼ L1

min�dW I

int;i

þ PL2i¼L1

min�dW II

int;i

þ PL2

i¼L1min�dW III

int;i

2 dWt

ext ¼ 0

ð24Þ

where dWtext 5 total external virtual work including external virtual

work dWext by service loadings and additional external virtual workdWpv

ext by Pv and zC 5 performance function of the constructionsystem,which is equal to zero.SubstitutingEq. (23) intoEq. (24) yields

z ¼ minðdWintÞ2 dWext ¼ PL2i¼ L1

min�dW I

int;i

þ PL2i¼L1

min�dW II

int;i

þ PL2

i¼L1min�dW III

int;i

2PL2

i¼ L1dWext ¼ dWpv

ext ¼PL2

i¼ L1pv;i � dD ð25Þ

Although Eq. (25) is set up based on the construction system thesolution of this equation can still reflect the safety status of a realdamaged frame structure. For example, if the structure is designedrobustly enough to sustain the initial column failure withoutcausing progressive collapse, a downward Pv is needed to push thedamaged frame to reach its ultimate state, and thus the value of z inEq. (25) will be positive. Otherwise, Pv has to be an upwardsupporting force vector with z being negative.

Minimum Internal Virtual Work

The structural members of a damaged frame are divided into threegroups; i.e., structural beams, directly connected columns, and

indirectly connected columns. According to Sucuo�glu et al. (1994),a significant change in themember axial forcewould primarily affectthe directly connected columns during progressive collapse of thedamaged frame. In contrast, variation of the axial force on othermembers (Fbc) is much smaller and thus can be regarded as lessimportant. Therefore, Fbc is treated as constant with its initialvalues obtained from the preliminary analysis of the intact frameunder service loadings. Adjustment of Fbc to consider its slightvariation during structural progressive collapse will be discussedsubsequently.

Structure Isolation

To calculate the minimum internal virtual work in the structuralperformance function, the ith story of the construction systemis divided into three parts (Parts a, b, and c) as shown inFig. 7(a). The working state of Part a is plotted in Fig. 7(b) whereFI;ic1, F

II;ic1 , F

III;ic1 , FI;i

c2, FII;ic2 , and FIII;i

c2 denote the axial forces on di-rectly connected columns; MI;i

2u;b1, MI;i2u;b2, M

I;i1u;c1, M

I;i2u;c2, M

II;i1u;b1,

MII;i1u;b2, M

II;i6 u;c1, M

II;i6 u;c2, M

III;i2u;b1, M

III;i2u;b2, M

III;i1u;c2, and MIII;i

2u;c1 rep-

resent the strength of the various end springs; Fil and Fi

r indicatethe vertical forces transferred from Parts b and c, respectively; andql;i, qr;i, and pv;i are two service loadings and virtual concentratedforce on the ith story, respectively. Among all these parameters arethe following:1. For each structural sample, ql;i and qr;i are given.2. From the vertical balance of Parts b and c, Fi

l and Fir can be

determined explicitly because the axial forces on the indirectlyconnected columns and beams ðFbcÞ are the currently availableconstants.

3. The functions of Fbc are MI;i2u;b1, M

I;i2u;b2, M

II;i1u;b1, M

II;i1u;b2,

MIII;i2u;b1, andM

III;i2u;b2 and are initially assumed to be the bending

strength of the corresponding structural beams.4. The Functions of FI;i

c1, FII;ic1 , F

III;ic1 , FI;i

c2, FII;ic2 , andF

III;ic2 areMI;i

1u;c1,MII;i

6 u;c1, MIII;i2u;c1, M

I;i2u;c2, M

II;i6 u;c2, and MIII;i

1u;c2, respectively,

where FI;ic2, F

II;ic2 , and FIII;i

c2 will be zero for the roof story.For any other story, FI;i

c2, FII;ic2 , and FIII;i

c2 can be obtained fromthe solution of the upper story.

Therefore, to calculate minimum internal virtual workminðdW I

int;iÞ, minðdW IIint;iÞ, and minðdW III

int;iÞ for the three criticalzones, only FI;i

c1, FII;ic1 , and FIII;i

c1 need to be determined by threeequations.

The first equation is built up based on the virtual work principle.Because the construction system is in its ultimate state, given a dD,the minimum internal virtual work within the isolated Part a,dWint;a, will be the sum of those produced by the involved threecritical zones

min�dWint;a

� ¼ min�dW I

int;i

þ min

�dW II

int;i

þ min

�dW III

int;i

¼ minnhMI;i

2 u;b1 þ MI;iþ u;c1

�FI;ic1

þ MI;i

2 u;c2

idD=ll;M

I;i2 u;b2dD=ll

oþ min

nhMII;i

þ u;b2 þ MII;iþ u;c1

�FII;ic1

þ MII;i

2 u;c2

idD=ll

þ MII;iþ u;b2dD=lr;M

II;iþ u;b1dD=ll þ MII;i

þ u;b2dD=lr;MII;iþ u;b1dD=ll þ

hMII;i

þ u;b1 þ MII;i2 u;c1

�FII;ic1

þ MII;i

þ u;c2

idD=lr

o

þ minnMIII;i

2 u;b1dD=lr;hMIII;i

2 u;b2 þ MIII;i2 u;c1

�FIII;ic1

þ MIII;i

þ u;c2

idD=lr ð26Þ

o

Fig. 6. Possible failure modes of Type III critical zone: (a) first possiblefailure mode; (b) second possible failure mode

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The external virtual work dWext;a in this part is calculated by

dWext;a ¼ 12ql;i � dD � ll þ 1

2qr;i � dD � lr þ FII;i

c2 � dD þ pv;i � dD2FII;ic1 � dD (27)

Equating Eq. (26) with Eq. (27) produces the first equation

minnh

MI;i2 u;b1 þ MI;i

þ u;c1

�FI;ic1

þ MI;i

2 u;c2

i.ll;M

I;i2 u;b2=ll

oþ min

nhMII;i

þ u;b2 þ MII;iþ u;c1

�FII;ic1

þ MII;i

2 u;c2

i.ll

þ MII;iþ u;b2=lr;M

II;iþ u;b1=ll þ MII;i

þ u;b2=lr;MII;iþ u;b1=ll þ

hMII;i

þ u;b1 þ MII;i2 u;c1

�FII;ic1

þ MII;i

þ u;c2

i.lro

þ minnMIII;i

2 u;b1=lr;hMIII;i

2 u;b2 þ MIII;i2 u;c1

�FIII;ic1

þ MIII;i

þ u;c2

i.lro

¼ 12ql;i � ll þ 1

2qr;i � lr þ FII;i

c2 þ pv;i2FII;ic1 ð28Þ

Fig. 7. Structural isolation and working state of the construction system: (a) isolated three parts; (b) working state of Part a; (c) separation of Part a intoParts a11 and a12; (d) separation of Part a into Parts a21 and a22

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The second equation can be built up from the vertical balance of Parta, and is expressed as

FI;ic1 þ FII;i

c1 þ FIII;ic1 ¼ FI;i

c2 þ FII;ic2 þ FIII;i

c2 þ Fil þ Fi

r

þ ql;r � ll þ qr;i � lr þ pv;i ð29Þ

To construct the third equation, the behavior of the Type II criticalzone is further inspected and, based on the relationship betweenMII

1u;b1 and MII1u;b2, there exist three possible cases; i.e.,

MII1u;b1 ,MII

1u;b2, MII1u;b1 .MII

1u;b2, and MII1u;b1 5MII

1u;b2.For the first case where MII

1u;b1 ,MII1u;b2, it can be deduced as�

MIIþ u;b2 þ MII

þ u;c1 þ MII2 u;c2

dD�ll þ MII

þ u;b2dD�lr

.MIIþ u;b1dD

�ll þ MII

þ u;b2dD�lr ð30Þ

Therefore, Eq. (20) can be simplified as

min�dW II

int

� ¼ minhMII

þ u;b1dD�ll þ MII

þ u;b2dD�lr;M

IIþ u;b1dD

�ll

þ�MII

þ u;b1 þ MII2 u;c1 þ MII

þ u;c2

dD�lri

ð31Þ

Eq. (31) indicates that for this case, only two possible failure modesremain. In any of these two modes, the ESPH, with its strength beingMII

1u;b1, will occur at the end section of Beam 1. Correspondingly, Parta canbe further separated into twoparts; i.e., Parts a11and a12 as shownin Fig. 7(c), whereV II;i

b1 is the shear force of the section. It is obvious thatone Type I critical zone is involved in Part a11 and the minimuminternal virtualworkminðdWint;a11ÞwithinPart a11canbecalculatedby

min�dWint;a11

� ¼ minnh

MI;i2 u;b1 þ MI;i

þ u;c1

�FI;ic1

þ MI;i

2 u;c2

idD�ll;M

I;i2 u;b2dD

�llo

ð32Þ

In addition, the external virtual work minðdWext;a11Þ for Part a11under dD is

dWext;a11 ¼ 12ql;i � dD � ll2V II;i

b1 � dD2MII;iu;þ b1 � dD

�ll

ð33Þ

Equating Eq. (32) with Eq. (33) produces

minnh

MI;i2u;b1 þ MI;i

þ u;c1

�FI;ic1

þ MI;i

2 u;c2

i;MI;i

2 u;b2

oþ MII;i

þ u;b1 ¼ 12ql;i � l2l 2V II;i

b1 � ll ð34Þ

To satisfy the vertical balance of Part a11, shear force V II;ib1 can be

determined by

V II;ib1 ¼ FI;i

c2 þ Fil þ ql;i � ll 2FI;i

c1 ð35Þ

Substituting Eq. (35) into Eq. (34), the third equation is finallyestablished as

minnh

MI;i2 u;b1 þ MI;i

þ u;c1

�FI;ic1

þ MI;i

2 u;c2

i;MI;i

2 u;b2

oþMII;i

þ u;b1 ¼�FI;ic1 2FI;i

c22Fil 2

12ql;i � ll

� ll

ð36Þ

For the second case where MII;i1u;b1 .MII;i

1u;b2, the third equation canbe developed by the same method as

minnMIII;i

2 u;b1;hMIII;i

2 u;b2 þ MIII;i2 u;c1

�FIII;ic1

þ MIII;i

þ u;c2

ioþMII;i

þ u;b2 ¼�FIII;ic1 2FIII;i

c2 2Fir 2

12qr;i � lr

� lr

ð37Þ

For the third case whereMII;i1u;b1 5MII;i

1u;b2, either Eq. (36) or (37) canbe adopted as the third equation because two ESPHswill occur at theend of both Beams 1 and 2 in the Type II critical zone to form thestructurally weakest collapse mechanism.

Solution of the Minimum Internal Virtual Work

BasedonEq. (28), (29), and (36) or (37), the solution ofFI;ic1,F

II;ic1 , and

FIII;ic1 is straightforward (Huang 2008) and leads to the determination

of MI;i1u;c1, M

II;i6 u;c1, and MIII;i

2u;c1 with the P2M curves of the cor-responding columns and subsequent evaluation of the minimuminternal virtual works produced by the three critical zones in thestory; i.e., minðdW I

int;iÞ, minðdW IIint;iÞ, and minðdW III

int;iÞ withEq. (19)e(21). The axial forces in Column 2s of the three criticalzones in the ði2 1Þth story; i.e., FI;i21

c2 , FII;i21c2 , and FIII;i21

c2 should beequal to FI;i

c1, FII;ic1 , and FIII;i

c1 , respectively. In addition, the strengthof the corresponding end springs on these columns (MI;i21

2u;c2,MII;i216 u;c2,

and MIII;i211u;c2 ) can be evaluated from the related columns’ P2M

curves and the minimum internal virtual work by every critical zoneand that by the weakest collapse mechanism of the constructionsystem are obtained.

Positions of Real Plastic Hinges

Although the structural lumped plasticity model adopted in thisstudy significantly reduces the computational work in structuralreliability assessment, errors may be induced in identifying thepositive RPHs on beams the Type II critical zones because themaximum positive moment along the beams in the damaged span

Fig. 8. Moment distribution along the damaged span of the con-struction system

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does not necessarily appear at the end section as shown in Fig. 8.These errors will finally cause the performance function and thereliability of the damaged frame to be overestimated.

To achieve a more accurate reliability assessment, the loca-tion of a positive RPH on beams in the Type II critical zonesneeds to be considered. One practical approach for this purposeis to employ reduced strength to the corresponding end spring.The equations to compute the reduced strength will be derivedsubsequently.

Positions of Real Plastic Hinges on Beams in the LeftDamaged Span

It is assumed that a positive RPH occurs on the beam of the ithstory in the left damaged span as circled in Fig. 8. The workingstate and moment diagram of the selected beam can be plotted inFig. 9. As the maximum positive moment reaches the beampositive strength MII;i

1ur;b1 at a location xl;i, a positive RPH will

appear there and apparently the moment at the right end sectionbecomes lesser thanMII;i

1ur;b1. To calculate xl;i, the beam is dividedinto two parts at xl;i.

Two equations are established based on the virtual work principleand the balance of the left part of the beam in Fig. 9(c), expressed,respectively, as

minnh

MI;i2 u;b1 þ MI;i

þ u;c1

�FI;ic1

þ MI;i

2 u;c2

i.xl;i;M

I;i2 u;b2=xl;i

oþ MII;i

þ ur;b1=xl;i ¼ 12� ql;i � xl;i ð38Þ

and

FI;ic1 ¼ FI;i

c2 þ Fil þ ql;i � xl;i ð39Þ

Substituting Eq. (39) into Eq. (38) produces

Fig. 9. Positions of the positive RPHs in the left and right damaged spans: (a) working state; (b) moment distribution; (c) left part of the beam;(d) working state; (e) moment distribution; (f) right part of the beam

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minnh

MI;i2 u;b1 þ MI;i

þ u;c1

�FI;ic2 þ Fi

l þ ql;i � xl;i

þMI;i2 u;c2

i;MI;i

2 u;b2

oþ MII;i

þ ur;b1 ¼ 12 � ql;i

�ql;i � xl;i

�2ð40Þ

By solving Eq. (40), xl;i can be obtained. Obviously, if xl;i $ ll,a positive RPHwill occur at the right end section and the strength ofthe corresponding end spring MII;i

1u;b1 can be considered as themember bending strength MII;i

1ur;b1. However, if xl;i # ll, a positiveRPH should be located at the calculated position of xl;i. In this case,themoment at the right end section of the beamcannot reachMII;i

1ur;b1.Therefore, reduced strengthmust be employed for the correspondingend spring MII;i

1u;b1 in the structural model. The calculation of thereduced strength can be carried out according to themoment diagramof the member in Fig. 9(b) to ensure that the maximum positivemoment of the select beam will not exceed its positive flexuralstrength MII;i

1ur;b1 and to assure that the accuracy of the performancefunction calculated based on the construction system will not beaffected by fixation of the springs at the end sections. Thus, thereduced strength for the corresponding end springMII;i

1u;b1 on Beam 1in the Type II critical zone can be obtained by

MII;iþ u;b1 ¼ MII;i

þ ur;b1 2 12ql;i �

�ll 2 xl;i

�2 ð41ÞFig. 10. Example beam

Fig. 11. Effectiveness of the reduced strength in considering the position of the RPHs

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Positions of Real Plastic Hinges on Beamsin the Right Damaged Span

A similar method can be applied to the beam within the rightdamaged span [see Fig. 9(f)]. The position of its positive RPH (xl; i)and the reduced strength of the corresponding end springMII;i

1u;b2 onBeam 2 in the Type II critical zone can be obtained, respectively, by

MII;iþur;b2 þ min

nMIII;i

2u;b1;hMIII;i

2u;b2 þ MIII;i2u;c1

��FIII;ic2 þ Fi

r þ qr;i � xr;iþ MIII;i

þu;c2

io¼ 1

2 � qr;i

�qr;i � xr;i

2ð42Þ

MII;iþ u;b2 ¼ MII;i

þ ur;b2 2 12qr;i �

�lr 2 xr;i

�2 ð43Þ

Numerical Example

To demonstrate the effectiveness of using reduced strength to ac-count for the position of a positive RPH, a simple example ofa beam that collapses under a uniformly distributed service loadingq of 50 kN×m is used here, as shown in Fig. 10. The positive andnegative strengths of the beam (i.e.,M1ur andM2ur) were assumedto be 225 and 400 kN×m, respectively. A construction system wascreated by applying a virtual concentrated force Pv at the left end.In addition, two springs were introduced on the constructionsystem to simulate the behavior of the plastic hinges. According tothe locations and strengths of the springs, three different models forcalculating external virtual work dWext done by q, internal virtual

work dWint, and performance function z in terms of dWint minusdWext were inspected, as depicted in Fig. 11 and 12.

The first model shown in Fig. 11 reflects the real collapsemechanism of the construction beam system. In this model, Spring 1with its strengthM1 is located at the positive RPH position with x55,000 mm and Spring 2 with its strengthM2 is fixed at the right endsection. Here, x is determined from Eq. (42) with MII;i

1ur;b2 5225 kN ×m, MIII;i

2u;b1 5 400 kN ×m, qr;i 5 50 kN×m, lr 5 6;000 mm,the other parameters are equal to infinity, and M1 and M2 areconsidered to be positive beam strength M1ur and negative beamstrength M2ur to simulate the positive and negative RPHs,respectively.

In the secondmodel, Spring 1 of strengthM1 is equal toM1ur andis fixed at the left end section of the beam to investigate the errorinduced by fixing the ESPH at the beam end section without con-sidering the position effect of the positive RPH. The only differencebetween the third and secondmodel is that a reduced strength of 200kN×m was employed for M1 in the third model to considerthe position of the positive RPH that was not at the end section.The value of the reduced strength is calculated from Eq. (43)with MII;i

1ur;b2 5 225 kN ×m, qr;i 5 50 kN×m, lr 5 6;000 mm, andxr;i 5 5;000 mm.

For dD5 1;000mm, external virtual work dWext done by serviceloading q, internal virtual work dWint, performance functionz5 dWint 2 dWext, as well as virtual concentrated force Pv 5 z wereevaluated for the aforementioned three models and are listed inFig. 11. Comparing the results obtained from the first and secondmodels shows that by fixing the springs at the beam’s end section,performance function z in the second model is overestimated byabout 8.4%. However, this error was compensated for by using thereduced strength for Spring 1 in the third model, where an accurateperformance function was attained.

Fig. 12. Models of critical zones in various failure modes

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DeterminationofVirtualConcentratedForceVectorPv

Because the construction system is established by applying a virtualconcentrated force vector Pv 5 fpv;igði5 L1⋯L2Þ on the damagedframe at the nodes above the failed column tomake the structure stayin its ultimate state, under dD, the external virtual work by Pv

denoted as dWpvext can be expressed as

dWpvext ¼ dD � pv;L1

PL2i¼L1

pv;i=pv;L1 ¼ minðdWintÞ2 dWext

ð44Þ

where dWext 5 virtualworkby service loadings. If the ratioofPv;i=Pv;L1

is predefined, the virtual force on ith story Pi can be calculated by

pv;i ¼ pv;L1 � pv;i=pv;L1 ð45Þ

where pv;L1 can be evaluated from Eq. (44), written as

pv;L1 ¼ ½minðdWintÞ2 dWext�.

dD � PL2i¼L1

pv;i=pv;L1

!ð46Þ

Based on Eq. (45) and (46), it can be observed that the magnitudeof concentrated virtual force fpv;ig depends on minimum internalvirtual work minðdWintÞ. On the other hand, Eq. (28) and (29) in-dicate that the axial forces on directly connected columns FI;i

c1, FII;ic1 ,

FIII;ic1 , and minimum internal virtual work minðdWintÞ of the con-

struction system are functions of the magnitude of fpv;ig. Thus, aniterative procedure is required. The flowchart of the iterative pro-cedure is shown in Fig. 13 and the process when combined with theMCS method is expressed as Fig. 14.

Axial Forces on Indirectly Connected Columns andBeams

A simplification was employed in the previous calculation of theminimum internal virtual work, where axial forces Fbc on the

Fig. 13. Flowchart for the calculation of the performance function

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indirectly connected columns and structural beams were treated asconstants with their initial values obtained from a preliminarystructural analysis. For further accuracy, modification of thestructure model was executed by following three steps for structuralanalysis.1. Pin connections were employed to replace the ESPHs accord-

ing to the failure mode of the critical zones.2. Bending moments equal to the strength of the related ESPHs

were applied on both sides of the pin connections.3. An additional vertical support was introduced at the top node

of the failed column as shown in Fig. 15 to avoid singularity ofthe structural stiffness matrix induced by the first and secondsteps. The reaction force on the structure by this additionalvertical support was zero and thus the axial forces on theindirectly connected columns and structural beams ðFbcÞwouldnot be affected by the third modification step.

Numerical Example

To demonstrate the application of the previous approach, theprogressive collapse occurrence possibility for a symmetric 3-bay,4-story RC frame structure was evaluated based on Fig. 13 and 14.

The structure was subjected to uniformly distributed serviceloading with the characteristic values of dead load (DL) and liveload (LL) as DLk 5 27 kN×m and LLk 5 30 kN×m. The charac-teristic compressive strength of concrete fck was 30 MPa, whilethe characteristic yield strength of reinforcing hot-rolled steelfyk was 400 MPa. The dimensions of the structural members arepresented in Fig. 16(a) with the cross-sectional data detailed inFig. 17. The beams within the same span and columns on the sameaxis had identical cross sections and reinforcements. Two types ofinitial column failure in the first level were checked and thecorresponding damaged frame structures are also plotted in Fig.16(b and c).

Basic Random Variables

According to the analysis conducted by Val et al. (1997a, b), thecompressive strength of concrete fc, yield strength of reinforcingsteel fy, and effective depth of structural members d have beenidentified as the major influencing parameters on structural re-liability. Alongwith service loadings DL and LL, they are defined asthe basic random variables.

According to the Comité Euro-International du Béton (CEB)(1993), the mean value of the compressive strength of concrete fcmcan be estimated by

fcm ¼ fck þ 8 ð47Þ

Considering fck as 0.05 fractile of fc, and fc is normally distributed,its coefficient of variation (COV) may be calculated by

COVfc ¼ 8=½1:64 � ðfck þ 8Þ� ð48Þ

For fy, COVfy is assumed to be 0.08 (MacGregor et al. 1983; Valet al. 1997a), and mean value fym can be obtained as

fym ¼ fyk.�

12 1:64 � COVfy

ð49Þ

The DL is considered to be a normal random variable while LL isrepresented by a Type 1 extreme value distribution. The COVs forthese two parameters (COVDL and COVLL) are 0.05 and 0.40,respectively, and their mean (DLm and LLm) are determined as inIsrael et al. (1987) and Val et al. (1997a, b)

Fig. 14. Flowchart for reliability assessment of the damaged structure

Fig. 15. Modified model of the construction system

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DLm ¼ 1:05 � DLk ð50Þ

and

LLm ¼ cLLLLk=ð1 þ 1:305 � COVLLÞ ð51Þ

where cLL 5 frequent value of LL used for a design situation as-sociated with the column failure in the detonation conditions, takenas 0.6 in this example.

According toMirza andMacGregor (1979), effective depths d ofvarious structural members are assumed to be normally distributedwith mean values equal to the nominal values and COVs being 0.05.In addition to the aforementioned five basic random variables, theuncertainties of concrete elastic modulus Ec and structural memberwidth b are also accounted for in this example. Considering the highdegree of correlation between Ec and fc, the former can be estimatedby Mirza et al. (1979) and Val et al. (1997b)

Ec ¼ aEð0:1fcÞ1=3 ð52Þ

where aE 5 elastic modulus coefficient of concrete, assumed to bean independent random variable. Based on previous research, the

statistic properties of all random variables are summarized inTable 1. Assuming each of the random variables is fully correlatedwithin a structural member and the variables are independent of eachother, there are 164 random variables for the example structure.Extensive studies and proper consideration need to be made on thecorrelation coefficients of various random variables in the reliabilityassessment of a real structure.

Results and Discussion

Setting the ratio pv;i=pv;L1 in Eq. (45) and (46) equal to 1, the oc-currence probabilities of damaged frame collapse events Pf werecalculated based on Fig. 13 and 14. To ensure attainment of con-vergence, various numbers Ns of structural samples were adopted inthe MCSs. The curves of Pf versus Ns as plotted in Fig. 18(a and b)show that Pf converges quickly with Ns increasing from 2,000 to80,000. When 80,000 structural samples were used in the MCS, theresultinged Pf were 0.22883 and 0.018288 for the two initialcolumn failure cases, respectively. It is obvious that loss of anexternal column is more critical for this particular frame than lossof an internal column. Because more members were involved ingenerating theweakest structural collapsemechanism after loss of an

Fig. 16. Four-story RC frame structure: (a) intact frame structure; (b) Case 1 (external column failure); (c) Case 2 (internal column failure)

Fig. 17. Dimension and reinforcement for various cross sections

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internal column, this would not only increase the mean value ofminðdWintÞ but reduce its COV. Furthermore, the COV of dWext wasreduced in the case of an internal column failure because the serviceloadings on more structural members were included in the calcu-lation of dWext and the beams in themiddle span of the structure weremore strongly reinforced than those in the side span.

In addition to Pf , the distributions of minðdWintÞ, dWext, and zalong with the mean values and COVs are plotted in Fig. 19(aef). Itcan be observed that the distributions of minðdWintÞ have a goodmatch with the normal probability density curves in both casesbecause all the random parameters are assumed to be normallydistributed. However, apparent deviations between the histogramsof dWext and the normal density curves can be observed as a resultof the Type I extreme value distribution for the LL.

The failure modes of the critical zones that occurred in thestructural weakest collapse mechanism were recorded for eachstructural sample. In addition, the total numbers of such occurrenceswere counted and are listed in Table 2 for both initial column losscases. It can be seen from Table 2 that the Type I, II, and III criticalzones of each story will fail mostly at the second, second, and firstmodes, respectively, for 80,000 samples. This observation indicatesthat the ESPHs of the structural weakest collapse mechanisms arepredominantly located on the beams in the damaged span. Only ina few samples, the Type II critical zones in the first and fourthstories (for Case 1) or the Type I critical zone in the fourth story (forCase 2) will fail in the mode with an ESPH appearing on the relatedcolumn, because there is only one structural column connected withone beam for the previous three critical zones. Although the columnwas intentionally designed to be stronger than the beam, there wouldbe a still smaller probability for the bending strength of the beam

being greater than that of the connected column within 80,000samples; this possibility further led to the ESPH appearing on thecolumn in a few samples. Nevertheless, it can be concluded based onTable 2 that the most effective way to improve the safety of thisframe structure against progressive collapse should be by increasingthe strength of the structural beams.

Summary and Conclusion

A structural performance function was established and the pro-cedure to solve this performance function was developed based ona construction system. The interactive effects between the strengthof the end springs and the axial forces on the related members wereconsidered in the calculation of minimum internal work. In addition,the reduced strengths were computed to account for the positions ofpositive RPHs that were not located on the beam end sections. Afterfixing virtual concentrated force vector Pv 5 fpv;ig and the axialforces on the indirectly connected columns and beams ðFbcÞ, thisprocedure became straightforward.

Modification of structural model was carried out based on theweakest collapsemechanism, while the axial forces on the indirectlyconnected columns and beams ðFbcÞ were updated according to therevisedmodel.Virtual concentrated force vectorPvwas also updatedbased on Eq. (45) and (46). The flowcharts for the calculatingstructural performance function and reliability assessment are shownin Fig. 13 and 14.

Implementations of the developed procedure were demonstratedfor a 4-story RC frame structure under two different cases of columnfailure with the incorporation of uncertainties in the material

Table 1. Distributions of Random Variables

Parameter

Variable

d b fc a fv DL LL

Distribution type Normal Normal Normal Normal Normal Normal Type I extreme valueMean value Nominal valuea Nominal valueb 38 2.15 3 1014 460 28.35 11.8COV 0.05 0.03 0.13 0.15 0.08 0.05 0.4aVariable d5 h2 c or d5 h2 c9; h and c are listed in Fig. 17.bVariable b is listed in Fig. 17.

Fig. 18. Collapse probability versus the number of structural samples: (a) Case 1 (external column failure); (b) Case 2 (internal column failure)

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properties, applied loadings, and structural geometries. The nu-merical results compared the failure probability of the structuresubjected to loss of an external column against that subjected to lossof an internal column. In addition, the results also show that theESPHs of the weakest collapse mechanism were predominantlylocated on the beams in the damaged span, which revealed the best

method to improve the structural safety by enhancing the structuralbeams. This paper presents the approach for reliability assessment ofa damaged RC frame structural system against progressive collapseunder static loading conditions. Extension of the developed ap-proach will be carried out subsequently to consider structural pro-gressive collapse under dynamic loading conditions.

Fig. 19.Distributions of minðdWintÞ, dWext, and z: (a) distributions ofminðdWintÞ in Case 1; (b) distributions of minðdWintÞ in Case 2; (c) distributionsof dWext in Case 1; (d) distributions of dWext in Case 2; (e) distributions of z in Case 1; (f) distributions of z in Case 2

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General Services Administration (GSA). (2000). Progressive collapseanalysis and design guidelines for new federal office buildings andmajor modernization projects, GSA, Washington, DC.

Huang, Z. W. (2008). “Reliability assessment of damaged ductile RC framestructures against progressive collapse in close-in detonation con-ditions.” Ph.D. thesis, Nanyang Technological Univ., Singapore.

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Sasani,M., Kazemi, A., Sagiroglu, S. G., and Forest, S. (2011). “Progressivecollapse of an actual 11-story structure subjected to severe initialdamage.” J. Struct. Eng., 137(9), 893e902.

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Table 2. Occurrence Numbers of Various Failure Modes of the Critical Zones

Case Critical zone

Occurrence numbers of critical zone failure modes (first mode/second mode/third modea)

Story 1 Story 2 Story 3 Story 4

1 Type II 0/79,968/32 0/80,000/0 0/80,000/0 0/79,999/1Type III 80,000/0/—a 80,000/0/—a 80,000/0/— 80,000/0/—a

2 Type I 0/80,000/—a 0/80,000/—a 0/80,000/—a 6,834/73,166/—a

Type II 0/80,000/0 0/80,000/0 0/80,000/0 0/80,000/0Type III 80,000/0/—a 80,000/0/—a 80,000/0/—a 80,000/0/—a

aThe third failure mode does not exist for the denoted critical zone.

JOURNAL OF ENGINEERING MECHANICS © ASCE / JANUARY 2013 / 17

J. Eng. Mech. 2013.139:1-17.

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