International Journal of High-Rise Buildings International Journal of June 2020, Vol 9, No 2, 127-154 High-Rise Buildings https://doi.org/10.21022/IJHRB.2020.9.2.127 www.ctbuh-korea.org/ijhrb/index.php

Review on Quantitative Measures of Robustness for Building Structures Against Disproportionate Collapse

Jian Jiang1, Qijie Zhang1, Liulian Li2, Wei Chen1, Jihong Ye1, Guo-Qiang Li3,† 1Jiangsu Key Laboratory of Environmental Impact and Structural Safety in Engineering, China University of Mining and Technology, Xuzhou 221116, China 2The First Construction Engineering Company Ltd. of China Construction Second Engineering Bureau, Beijing 100176, China 3State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China

Abstract

Disproportionate collapse triggered by local structural failure may cause huge casualties and economic losses, being one of the most critical civil engineering incidents. It is generally recognized that ensuring robustness of a structure, defined as its insensitivity to local failure, is the most acceptable and effective method to arrest disproportionate collapse. To date, the concept of robustness in its definition and quantification is still an issue of controversy. This paper presents a detailed review on about 50 quantitative measures of robustness for building structures, being classified into structural attribute-based and structural performance-based measures (deterministic and probabilistic). The definition of robustness is first described and distinguished from that of collapse resistance, vulnerability and redundancy. The review shows that deterministic measures predominate in quantifying structural robustness by comparing the structural responses of an intact and damaged structure. The attribute-based measures based on structural topology and stiffness are only applicable to elastic state of simple structural forms while the probabilistic measures receive growing interest by accounting for uncertainties in abnormal events, local failure, structural system and failure-induced consequences, which can be used for decision-making tools. There is still a lack of generalized quantifications of robustness, which should be derived based on the definition and design objectives and on the response of a structure to local damage as well as the associated consequences of collapse. Critical issues and recommendations for future design and research on quantification of robustness are provided from the views of column removal scenarios, types of structures, regularity of structural layouts, collapse modes, numerical methods, multiple hazards, degrees of robustness, partial damage of components, acceptable design criteria.

Keywords: Robustness, Disproportionate collapse, Quantitative measure, Review, Deterministic, Probabilistic

1. Introduction and Yoo 2019). Therefore, it is of great importance to understand the collapse behavior of tall structures, and The study of disproportionate collapse under extreme moreover to develop reliable and efficient methodologies or accidental loads can trace back to the 1940s, when to arrest collapse. It is generally recognized that increasing Baker (1948) studied how buildings collapsed under robustness is the most acceptable and effective method. bombing in London during the Second World War. Past All buildings are vulnerable to disproportionate collapse design and research work for disproportionate and/or in varying degrees (Ellingwood and Leyendecker 1978; progressive collapse of structures has proceeded in waves Nair 2006; NIST 2007; Kwon et al. 2012), and it has been initiated in the aftermath of best-known events, parti- widely accepted that it is not safe to assume that a cularly the Ronan Point failure 1968 (Pearson and Delatte structure designed for normal conditions will withstand 2005), Murrah Federal Building bombing 1995 (Corley et abnormal or accidental conditions (Ha et al. 2017). Therefore, al. 1998) and the World Trade Center collapse 2001 (NIST in the last 20 years, great advances in design and research 2008). Disproportionate collapse is a relatively rare event have been made on collapse resistance of building structures, as it requires both an abnormal load to initiate a local supported by advanced computational tools. A detailed damage and a structure that lacks adequate continuity, comparison of available design codes, standards and guides ductility and redundancy to resist the spread of failure. (GSA 2005; ASCE 7 2005; JSSC 2005; DoD 2013; EN However, for high-rise building structures (Chung et al. 1991-1-7 2010) can be found in references (Dusenberry 2016; Zhou and Liu 2019), the consequence and hence 2002; Ellingwood and Dusenberry 2005; Ellingwood 2006; the risk of collapse is immense (Li et al. 2018a; Chung Mohamed 2006; Nair 2006; Starossek 2006; Stevens et al. 2011; Marchand and Stevens 2015; Russell et al. 2019). †Corresponding author: Guo-Qiang Li The comparison highlights an apparent discrepancy between Tel: +86-21-6598-2975, Fax: +86-21-6598-3431 the observed structural disproportionate collapse of buildings E-mail: [email protected] and current guidelines to mitigate such outcomes. Research 128 Jian Jiang et al. | International Journal of High-Rise Buildings advances have been summarized extensively in review vulnerability and redundancy. A classification of the papers (Starossek 2007; Agarwal et al. 2012; El-Tawil et al. existing quantitative measures of robustness was presented 2013; Byfield et al. 2014; Qian and Li 2015; Adam et al. in Section 3. A detailed overview was given on the classi- 2018; Jiang and Li 2018; Kunnath et al. 2018; Abdelwahed fied structural attribute-based measures in Secton 4, and 2019; Bita et al. 2019; Russell et al. 2019; Subki et al. structural performance-based deterministic and probabilistic 2019; Stochino et al 2019; Azim et al. 2020; Kiakojouri et measures in Section 5. Finally, the main issues and recom- al. 2020), and academic books (Starossek 2009, 2018; Fu mendations are provided in Section 6. This paper focused 2016; Isobe 2017). on steel framed structures, however, noting that similar To defend buildings against unpredictable accidental findings are also applicable to reinforced concrete structures. events, the general consensus in the structural engineering community is to ensure robustness so that spread of the 2. Definition of Disproportionate Collapse initial damage is controlled and disproportionate collapse and Robustness is ultimately prevented. A quantitative measure would be useful to examine the performance of a structure in terms 2.1. Disproportionate Collapse vs Progressive Collapse of its robustness. The growing acceptance of performance- It is necessary to first distinguish between “dispro- based design is accompanied by a need for evaluation, portionate collapse” and “progressive collapse” (Adam et optimization and regulation of robustness under a set of al. 2018), as illustrated in Figure 1. The former concept extreme natural and man-made events. To achieve these was first described in the UK code (Building Regulation tasks, robustness measures must meet the requirements of 2010) as “the building shall be constructed so that in the being expressive, objective, simple, calculable, and generally event of an accident the building will not suffer collapse applicable (Starossek and Haberland 2011), which cannot to an extent disproportionate to the cause”, while the all be satisfied to the same level at the same time. To date, latter was first systematically discussed in the US design the concept of robust structures is still an issue of con- guidelines (GSA 2005) as defined in ASCE 7 (2005) “the troversy since there are no well established and generally spread of an initial local failure from element to element, accepted criteria for a consistent definition of robustness, eventually resulting in the collapse of an entire structure which hinders development of quantitative measures of or a disproportionately large part of it”. A list of existing structural robustness. There are some review papers related definitions of disproportionate/progressive collapse is to quantification of structural robustness (Ghosn et al. shown in Appendix A. No matter what is the definition of 2016; Zio 2016; Adam et al. 2018). However, they focus disproportionate or progressive collapse, it indicates that on eigher structural members, indicators (qualitative concept) large displacements, even failure of individual structural of robustness, or a broad subject of civil engineering and members are acceptable provided that structural collapse infrastructure networks, but are not directly applicable for is prevented. quantifying robustness of buildings which is the objective These two terms are often, mistakenly, used to be of this paper. synonymous. This is partly because disproportionate collapse This paper presented a comprehensive review on quanti- often occurs in a progressive manner, and progressive tative measures (rather than indicator) of robustness for collapse can be disproportionate. However, a collapse buildings against disproportionate collapse, especially may be progressive in nature but not necessarily dispro- focusing on steel framed buildings. It was started by portionate in its extents, for example if a collapse is recalling the definition of robustness (Section 2) and its arrested after it progresses through a limited extent of difference from similar terms such as collapse resistance, structural bays. Vice versa, a collapse may be dispro-

Figure 1. Relation between Disproportionate, Progressive Figure 2. Relation between Robustness and Collapse Resistance, and General Collapse. Vulnerability, Redundancy. Review on Quantitative Measures of Robustness for Building Structures Against Disproportionate Collapse 129 portionate but not necessarily progressive if that “robustness is broadly recognized to be a property the collapse is confined to a single but large structural bay. which can not only be associated with the structure itself The Murrah Building and WTC 1 and 2 are progressive but must be considered as a product of several indicators: collapse but cannot be reasonably labeled as risk, redundancy, ductility, consequences of structural disproportionate collapse since the initial explosion or component and system failures, variability of loads and impact is also very large (Nair 2006). In other words, a resistances, dependency of failure modes, performance of disproportionate collapse is only a judgement on the structural joints, occurrence probabilities of extraordinary consequences of the damage, while a progressive collapse loads and environmental exposures, strategies for structural describes the characteristics and mechanism of collapse monitoring and maintenance, emergency preparedness and behavior. Therefore, dispro-portionate collapse is more evacuation plans and general structural coherence”. appropriate in the context of design to accommodate By summarizing these definitions, robustness is generally specified design objectives, while progressive collapse is defined as insensitivity of a structure to its initial damage more suitable in the context of research when referring to or local failure, i.e. a structure is robust if an initial the physical phenomenon and mechanism of collapse. damage does not lead to disproportionate collapse. In the Moreover, disproportionate collapse must not be confused literature, a large number of similar terms with various with general or global collapse due to strong earthquake, meanings are used associated with robustness, such as wind, hurricane, etc. The former is usually with respect to collapse resistance, vulnerability, redundancy, reliability, abnormal loads such as blast, impact, and fire where the integrity, damage tolerance, resilience, stability, ductility, initial damage is localized, while the latter involves a toughness, susceptibility, fragility, etc. (Starossek et al. 2011). global series of simultaneous failure. There are some A distinction between robustness and the first three terms exceptions that earthquake may cause the damage of (the most confusing ones) is presented in this section to several corner columns of a building, and thus there is some help understand the meaning of robustness, and to provide a overlap between general collaspe and disproportionate/ basis for various quantifications as presented in the next progressive collapse as shown in Figure 1. The term sections. “disproportionate collapse” is used herein to facilitate a review on design-oriented application of robustness. 2.2.1. Robustness vs Collapse Resistance Robustness is associated with the probability use [ ], i.e. 2.2. Definition of Robustness P[CollapseD] and collapse resistance (the short form for The term “robustness” has been well-defined in various resistance against disproportionate collapse) with the fields beyond structural engineering (e.g., software engineering, probability use [ ] (Starossek and Haberland 2010), as quality control, design optimization, medicine, social sciences shown in Figure 3 (H is the abnormal event; D is local or finances). For example, robustness in quality control is damage; P[] is probability). This indicates that collapse defines as the degree to which a system is insensitive to resistance can be influenced in various ways, and one the effects that are not considered in design. A robust possibility is through the structural robustness. In other statistical technique is insensitive to small deviations in words, a robust structure is collapse resistant but not vice- the assumptions. In optimization theory, a robust system versa. Robustness is a property of the structure alone and is that whose performance is insensitive to uncertainties or independent of the possible causes and probabilities of random perturbations in design parameters. The awareness the initial local failure, while collapse resistance is a of the significance of robustness in structural engineering property dependent on both structural features and the has intensified over the years, but robustness is still an possible causes of initial local failure. A robust structure issue of controversy for its definition and quantification is collapse resistant because an initial damage does not despite substantial research and useful recommendations. spread dispropor-tionately, being achieved by alternative This poses difficulties with regard to interpretation as load paths or segmentation (isolation of a failure in a segment). well as regulation (Faber 2006). A non-robust structure can be made collapse resistant by A list of available definitions of robustness in design reducing the vulnerability of structural components to codes and research studies is provided in Appendix B. A weaken or prevent initial damage. A disproportionate definition is given in EN 1991-1-7 (2010) as “the ability collapse can be prevented by providing robustness (system of a structure to withstand events like fire, explosions, behavior) or by reducing the exposure to abnormal impact or the consequences of human error, without being actions (event control). This is in contrast to a broader damaged to an extent disproportionate to the original definition of robustness as given in EN 1991-1-7 (2010), cause”. A similar definition is provided in the U.S. General where robustness is referred to insensitivity to abnormal Services Administration (GSA) that describes robustness events rather than initial damage. In a word, the collapse as the ability of the structure to resist failure “due to its resistance is a broader concept, involving robustness of vigorous strength and toughness”. The workshop on the global structure, resistance of local components and robustness of structures held by Joint Committee on hazard events. Structural Safety (JCSS) in 2005 (Faber 2006) concluded 130 Jian Jiang et al. | International Journal of High-Rise Buildings

either abnormal events or an initial damage without disproportionate collapse. Before quantification, it is necessary to first distinguish three common terms: indicator, measure, index. An indicator describes the satisfication or not to certain conditions and circumstances, which is a qualitative or quantitative description of a property. The other two are used for quantitative description of a Figure 3. Definitions in the Context of Disproportionate Collapse. property. An index is a ratio for the change of values of a quantity, used for quantitative description. In a word, an 2.2.2. Robustness vs Vulnerability indicator is the most general form used to describe a Basically, vulnerability is not the antonym of robustness. property, and each measure can also be regarded as an Vulnerability decribes the sensitivity of a member or a indicator and an index is a special form of a measure. structure to damage events, rather than induced local Measures of robustness should be linked to a series of damage as used in robustness. On one hand, robustness is requirements such as expressiveness, objectivity, simplicity, related to system behavior, while vulnerability refers to calculability, and generality (Lind 1995; Starossek 2018). either member or system behavior (Starossek and Haberland These requirements are partly in conflict with each other, 2010), as shown in Figure 3. On the other hand, robustness and it may not be possible to meet them all to the same is a quality of the structural system alone, and is level at the same time. For example, it is possible to achieve independent of the cause of the damage, while vulnera- strong expressiveness but at the cost of calculability. bility is used for evaluating the consequences of a given Therefore, the above requirements may have to be limited. hazard (typically floor damage extent, economic losses or There are other requirements for a quantitative measure of number of fatalities), considering the type of action and the robustness, such as that it should be a decision-making tool structural response to that action (Felipe et al. 2018). The for design or redesign, valid to actually express the vulnera-bility of a structure will vary between different tolerance of damage, reliable to distinbuish different hazards, e.g. a structure may be vulnerable to vehicle damages among different systems, and reproducible. Any impact but not to seismic loading. However, the candidate for a measure of robustness should be assessed robustness of structures should work for both hazards. against these requirements. To quantify robustness it is necessary to consider possible 2.2.3. Robustness vs Redundancy scenarios of abnormal events, structural collapse, and Structural robustness the robustness and redundancy collapse-induced consequences. However, there is no are often used inter-changeably as synonymous terms. general rule to quantify the robustness of structures. However, they indicate different properties of a structural Starossek and Haberland (2008) divided the quanti- system. Structural redundancy is defined as the capability fications into structural attribute-based and structural of the structural system to redistribute the loads among its behavior-based measures. Sørensen (2011) and Chen et members through alternate loading path. In general, al. (2016b) divided the robustness measures of structures redundancy tends to enable a more robust structure and to divid into three categories with decreasing complexity: ensure that alternate load paths are available by means of risk-based, reliability or probability-based and deterministic structural ties, strength and ductility. This means structural performance-based measures. Adam et al. (2018) classified robustness strongly depends on redundancy. Alternatively, if the quantificaiton of robustness into threat-dependent and alternate paths are not available, the robustness of structues threat-independent methods. The former included all can also be ensured by introducing a discontinuity in the reliability/risk-based measures and some deterministic structure (segmentation) or by designing some critical measures. The structural attribute-based measures belonged elements to resist the extreme event (key element design). to threat-independent group. Lin et al. (2019) classified In this context, continuity (tie strength), redundancy robustness assessment methods into probabilistic and (alternate load path), ductility, segmentation, energy deterministic theory. The probabilistic methods were absorption capacity are identified as the means of accom- further divided into probability redundancy-based and plishing robustness. In addition, structure redundancy is vulnerability-based methods. related to a specific and prospective damage scenaro, In this paper, the existing quantitative methods were while robustness is a comprehensive measure based on a classified into two groups: structural attribute-based series of possible damage scenarios. The lower the structural measures and structural performance-based measures, as redundancy, the worse the robustness of a structure. shown in Figure 4. The former is evaluated based on the structural topology (geometry, configuration, connectivity 3. Classification of Quantitative Measures of of structures) and structural stiffness. Based on whether Robustness considering the load, it can be further classified into two categories: change to attribute alone and change to attribute Robustness refers to the ability of a structure to resist to and load. The structural performance-based methods Review on Quantitative Measures of Robustness for Building Structures Against Disproportionate Collapse 131

Figure 4. Classification of the Existing Measures of Structural Robustness. involve structural behaviour such as bearing capacity, system vulnerability focusing only on the structural ductility, redundancy, energy dissipation, probability of topology and stiffness. The topology of a structure was failure, risk, reliablity, which can be further divided into represented by a hierarchical model of clusers as a deterministic and probabilistic measures. The deterministic container of mimimum resisting substructures (“structural measures can be further divided into bearing capacity, ring” or “structural round”). The sequence of grouping deformation, damage extent, and energy-based measures. members into clusters was determined by ensuring an They focus on the responses of the system against increasing “well-formedness” of the target cluster. The collapse, without considering the randomness in abnormal well-formedness Q of a structural round in Eq. (1-1) is a events, system properties and consequences. In contrast, measure of the quality of the structural form, which is the the probabilistic measures can be further classified into mean value of the well-formedness qi of each node in the failure probability-based measures (involving randomness round as expressed in Eq. (1-2) or Eq. (1-3). of abnormal events and structural system), risk-based 1 Q = ---- q measures (involving randomness of abnormal events N∑ i (1-1) (causes) and consequences) and reliability-based measures (involving randomness of structural system alone). qi = det(kii) (for a static system) (1-2)

ω2 4. Review on Structural Attribute-based qi = det(kii + n Mii) (for a dynamic system) (1-3) Quantitative Robustness Measures where Q and qi is the well-formedness of a structural The structural attributes include structural topology and round and a node in the round, respectively; N is the total structural stiffness. The former represents the configu- number of the connections in the round; Kii is and Mii is ration or hierarchical model of structural components and the stiffness and mass sub-matrix associated with any their connectivity, while the latter involves important node i in the round, respectively; ωn is the this superscript information of the structure such as geometry dimension, natural frequency of the structure; det() means deter- member sizes, material properties, etc. The stiffness minant calculation. matrix can be easily computed for simple structures and This concept of well-formedness was used to identify its properties can be used to indicate the failure behavior failure scenarios (sequence of damage events) by three of structural systems. Based on whether the loading quantitative measures: separateness γ in Eq. (2-1) (ratio of condition was considered in the quantification, the structural well-formedness loss of the damaged structure to well- attribute-based measures of robustness were further formedness of the intact structure), relative damage divided into attribute alone-based and attributed and load- demand Dr in Eq. (2-2) (ratio of damage demand D of a based measures, which are presented in the following failure scenario to the maximum possible damage subsections. demand Dmax of the system), vulnerability index ϕ in Eq. (2-3) (ratio of separateness to relative damage demand). 4.1. Attribute Alone-based Robustness Index The damage demand is a measure of the effort required to A structural vulnerability theory was proposed by cause damage, which is proportional to the loss of structural researchers from University of Bristol (Agarwal et al. stiffness due to a damage. The separateness is a measure 2001, 2003; Blockley 2002; Pinto et al. 2002), aiming at of failure consequence, with a zero value representing no indentifying the possible failure modes and assessing the collapse and a unity value representing total collapse. 132 Jian Jiang et al. | International Journal of High-Rise Buildings

While the vulnerability index is a measure of the dispro- on the loading path and structural stiffness distribution. portionateness of the consequences (γ) to the damage (D). However, a simple sum of these internal forces lacks theoretical basis. QS()–QS() ′ γ = ------(2-1) QS() Ba= TK–1aKn (3-1) D D = ------n r (2-2) ()1–B = r D ∑ ii (3-2) max i=1 γ ϕ = ------where a is the transition matrix between member D (2-3) r internal forces and external forces; K is structural stiffness matrix; K'' is the member stiffness matrix; B is a where Q(S) and QS() ′ is the well-formedness of the transition matrix between member deformation terms and intact structure S and the damaged structure S', res- nondeformation terms; Bii is the term on the diagonalline pectively. of matrix B, i.e. the importance coefficient of the member Among a large number of possible failure scenarios, i; r is the redundancy factor. five important types of failure scenarios were figured out Hu (2007) assessed the structural stiffness through such as total, maximum, minimum, minimum demand fundamental frequency of a structure, and used the ratio and specific (or interested) failure scenarios. The total of stiffness degradation after member removal as the failure scenario requires the least effort to cause the total measure index. However, use of fundamental frequence collapse of the structure, and the maximum failure cannot account for other modal frequencies and the scenario has the maximum damage consequence from the sequence change of structural frequencies after damage. relative least effort. The minimum demand scenario has In addition, the member contributes less to the fun- the damage event easiest to cause any damage. The damental frequency may have great contribution to specific failure scenario is the one of designer’s interest. structural safety. The total failure scenario has a separateness equal to one Nafday (2008, 2011) proposed two redundancy factors and the highest vulnerability index, while the maximum denoted as system integrity distance matric δs (or system failure scenario is the one having highest vulnerability safety performance metric) in Eq. (4-1) and system index. integrity volume metric Δs (or degree of linear depen- Basically, this structural vulnerability theory is applicable dency in stiffness matrix) in Eq. (4-2). The former was to linear elastic analysis of framed structures, and can be defined as the reciprocal of the condition number κ(K) of used to find different levels of failure scenarios (total, the (n×n) stiffness matrix K, representing the shortest maximum, minimun failure scenarios) and the key elements distance from the stiffness matric (initial state) to set of of structures. Further work is needed to extend this theory singular matrices (limit state). This is based on the concept for other types of structures and plastic deformation. Ye that a singular stiffness matrix represents an unstable and Jiang (2018) improved this theory by considering the structure and it is desirable to have the stiffness matrix transition from rigid connection to pinned connection “far way” from the set of noninvertible singular matrices. through defining a rigid connection as a basic element in The δs ranges between 0 and 1, with a higher value the traditional vulnerability theory (only components are indicating more stable system. The Δs was defined as the treated as basic element). It was found that the improved determinant of normalized stiffness matrix KN, ranging method can accurately predict the damage location and between 0 and 1 (higher value means higher robustness). collapse mode, which cannot be achieved by the traditional The singularity of a stiffness matrix increased as the degree method. of linear dependency of its row or column vectors A similar method was proposed by Liu and Liu (2005) increased. and Gao and Liu (2008), based on the the theory of An importance measure I was also proposed by Nafday minimum of potential energy and structural stiffness. A (2008) to represent the contribution of a structural series of unit equivalent axial force, shear force and member to system safety, as expressed in Eq. (4-3). It was moment were imposed at the ends of individual structural defined as the ratio of the determinant of normalized member, and the sum of the induced axial force, shear stiffness matrix of the intact structure KN to the damaged K* force and moment in the member was used to measure the structure N , ranging from 1 to infinity. The higher the importance of the member. A component-level coefficient of importance measure, the more critical the member for importance Bii is expressed in Eq. (3). This method was survival of the structure. A similar member consequence Ci proposed for truss structures and was applicable for factor f was proposed by Nafday (2011) to search for elastic conditions, which is its limitation. The lower the key members, as shown in Eq. (4-4). The factor ranges importance coefficient, the higher the level of redundancy, from 0 to 1 with a lower value means a more critical and thus the better the robustness. This method depends member. Review on Quantitative Measures of Robustness for Building Structures Against Disproportionate Collapse 133

δ ==1/κ()K ,δ nK⁄ K–1 S S England et al. (2008) exteneded Eq. (1) for specific (system integrity distance matric) (4-1) loading conditions. The potential for damage propagation was examined through a new measure of hazard potential ΔSK= N (system integrity volume metric) (4-2) H in Eq. (6). Hazard potential is a measure of the potential for the progression of damage through a particular failure K scenario under a set of loads. For a set of failure scenarios I = ------N (Importance measure) (4-3) in a structure, the lowest hazard potential denotes the K* N most vulnerable failure scenario.

Ci = Ki ⁄ K U /U f N N (member consequence factor) (4-4) H = ------i 0- i F /F (6) i 0 Where κ is the condition number of a matrix; . Ki represents Euclidean matrix norm; N is the normalized Where Ui and Fi represent strain energy and well- stiffness matrix of the damaged structure after removal of formedness of a structure after the ith event, respectively; th the i member. U0 and F0 correspond to the undamaged state. Starossek and Haberland (2011) suggested a similar Ye et al. (2010) used generalized structural stiffness to stiffness-based measure Rs of robustness in Eq. (5) by measure the importance of members. The generalized comparing the stiffness matrix K0 of the intact structure structural stiffness Kstru can be calculated from the system and that Kj of the damaged structure after removing a internal energy U given a generalized loading force Fmax, structural element or a connection j. This measure can be as given in Eq. (7-1). The measure index I was defined in normalized to be in a range between 0 and 1, where a Eq. (7-2) as the ratio of damage-induced loss of generalized value of one represents the maximum possible robustness, structural stiffness to that of the intact structure. This while zero corresponds to a total lack of robustness. This index has a clear physical meaning for elastic behavior of index has a low level of expressiveness. Furthermore, the structures that the importance of a member is measured reduction in load-bearing capacity due to member by its contribution to generalized structural stiffness. Note removal is not well correlated with Rs, and thus Rs should that the generalized structural stiffness depends on both be regarded as a connectivity measure instead of a the structural topology and load. complete robustness measure. This is because system 1 1 connectivity (redundancy) provides only a partial con- K = ---F2 ---- (7-1) tribution to robustness. stru 2 maxU

det()K K –K R = min ------j - (5) I = ------stru,0 stru,- f S jdet()K K (7-2) 0 stru,0

Generally, the above topology and stiffness-based Where Kstru,0 and Kstru,f is the generalized structural vulnerability methods have the advantages of simplicity stiffness of the intact and damaged structure, respectively. and ease of calculation. However, they are applicable to The index I ranges between 0 and 1. The larger the index, identify weaknesses in a structure at an early stage in the the more important the member. design process (elastic analysis), rather than being a The above structural attribute-based methods (with and substitute for a full dynamic analysis under extreme without load) consider robustness as a fixed property of actions. The matrix determinant-based method lacks clear the system in terms of structural configuration (topology) physical meaning. Further work is needed to extend them and stiffness. These stiffness-based indices are applicable to account for elastoplastic behaviour and dynamic effects to the performance objective of stability (rather than bearing due to sudden failure. capacity or deformation limit), and and are more applicable for for an indicator rather than a measure for structural 4.2. Attribute and Load-based Robustness Index robustness since the algebraic properties of structural The effective load path depends on not only the distri- stiffness matrix lack clear physical meaning, which is bution of geometry and stiffness, but loading action. The difficult to be used in practical design. Most attribute- above methods in Section 4.1 fail to consider the effect of based methods deal with individual members and their member failure on the load redistribution path. In fact, the impact on the performance of the system. They are structural stiffness matrix used in the robustness measure- straightforward for simple structures like truss structures. ment excluding load is not the effective stiffness of a However, in dealing with complex structural system, structure to resist the load, where a member contributing different aspects should be considered including location less to the loading path may have a higher importance of each individual component, safety level of each than that contributing more to loading path. Therefore, member, stiffness sharing of each member, and material attribute and load-based methods have been proposed. behavior of each member, etc. In this case, the quantifi- 134 Jian Jiang et al. | International Journal of High-Rise Buildings cation of the importance of all components and ranking LF LF LF R ==------u-,R ------f,R =------f them in a proper order should be addressed on a system u LF f LF d LF (8) performance perspective but not on individual component 1 1 1 checking. Where Ru is the system reserve ratio for the ultimate limit state of the intact structure; Rf is the system reserve 5. Review on Structural Performance-based ratio for the functionality limit state of the intact Quantitative Robustness Measures structure; Rd is the system reserve ratio for the ultimate limit state of the damaged structure; LF , LF , LF , are 5.1. Deterministic Robustness Index u f d load factors of these three cases, defined as the ratio of Deterministic structural performance-based methods system capacity to the applied load; LF is the load ratio define robustness as the ratio of some specific structural 1 of the most critical member. properties (capacity, deformation, damage or energy) of Similar redundant factors were proposed in Eq. (9) by undamaged (or intact) and damaged structures. They are Feng and Moses (1986), Frangopol and Curley (1987) as: physically meaningful and convenient to calculate. However, the inherent randomness involved in the structural properties (1) Reserve redundant factor: and external loads cannot be reflected, which may greatly L R = ------intact affect the robustness of the structures. The following 1 L (9-1) subsections present the robustness measures based on design bearing capacity, deformation, damage extent and energy (2) Residual redundant factor: flow. L R = ------damage 2 L (9-2) 5.1.1. Bearing Capacity-based Robustness Index intact Based on redundancy and robustness of bridge structures (Frangopol and Nakib 1991; Ghosn and Moses 1998; (3) Strength redundant factor: Wisniewski et al. 2006; Ghosn et al. 2010), three deter- L R = ------intact - ministic capacity-based measures of robustness were pro- 3 L –L (9-3) posed in Eq. (8). The robustness or redundancy was defined intact damage as the capability of a structure to continue to carry loads after the failure of the most critical member. The load Where Lintact is the collapse load of the intact structure; factors LFi provide deterministic estimates of critical Ldesign is the design load; Ldamage is the collapse load of the limit states that describe the safety of a structural system, damaged structure. as shown in Figure 5. These load multipliers were usually It is interesting to note that the product of R2 and R3 obtained by performing an incremental nonlinear finite indicates whether the damaged structure will survive the element analysis. The overall safety of the system was design load (i.e. design load survivability, and survival assessed by a redundancy factor, defined as the minimum implies R2R3>1). of the three redundancy ratios. The proposed metho- L dology is a combination of partial safety factor method R R = ------damage 2 3 L (9-4) and nonlinear static analysis. design

Although the reserve redundant factor in Eq. (9-1) and design load survivability in Eq. (9-4) are descriptive of the collapse capacity relative to the design load, they do not quantify robustness by comparing the capacity of the undamaged and damaged structure as required by the definition of robustness. The residual redundant factor in Eq. (9-2) and strength redundant factor in Eq. (9-3) makes this comparison, but is not helpful for design purpose because it does not incorporate the design load. Fallon et al. (2016) proposed a new relative robustness index (RRI) to relate the capacity of the damaged structure to that of the intact structure and design load as shown in Eq. (10). A negative RRI indicates that the damaged structure does not meet the design load requirement. A positive RRI between 0 and 1 indicates that the capacity of the damaged Figure 5. Definitions of Capacities of an Intact and Damaged structure exceeds the design load but is less than the Structure Under Different Limit States. capacity of the undamaged structure. A value of 1 indicates Review on Quantitative Measures of Robustness for Building Structures Against Disproportionate Collapse 135

GR no loss of capacity because of the local damage. Eq. (10) GNR = ------j - j max()GR ,GR ,,… GR (13-2) offers a distinct improvement over the expressions in Eq. 1 2 ne (9) and can be easily incorporated into a uniform pushdown evaluation for progressive collapse resistance. In addition, Where GRj is the generalized redundancy of the structure th the consequence of collapse was quantified in terms of for j damage; GNRj is the normalized vertion of GRj; Sij the ratio of damaged floor area to the total area of the is the response sensitivity of the ith element for jth damage; structure, making the method available for practical V is the total volume of the structure; Vi is the volume of design. the ith element; ne is the number of elements in the structure. L Similar to the above residual redundant factor in Eq. (9------damaged-–1 L –L L λ –1 2), Khandelwal and El-Tawil (2011) proposed an overload RRI===------damaged design ------dasign ------damaged - L –L L λ –1 factor, defined as the ratio of failure load to the nominal intact design ------intact –1 undamaged gravity load of a damaged structure. The overload factor L design determined from pushdown analyses with different load (10) imposing schemes together with different collapse modes were used to measure robustness of structures. The above methods consider the system robustness by Coefficients of importance of components (or member comparing the resistance of damaged and intact structures sensitivity index) were proposed (Gao 2009; Huang and and design load. To examine the relationship between Li 2012; Choi and Chang 2009) to evaluate the robustness member capacity and system capacity, Hendawi and of structures. The coefficient of importance is defined as: Frangopol (1994) proposed a system safety factor SSF as: λ –λ R –R η = ------0 i γ = ------0 i i λ or i R (14) ∑i Ri 0 0 SSF= ------(11) P Where ηi or γi is the coefficient of importance; λ0 is the capacity ratio of the intact structure, defined as the ratio Where Ri is the mean resistance of the member i, and of the ultimate loading capacity to the applied load; λi is P is the mean of the total applied load. the capacity ratio of the damaged structure due to the To consider the probability of failure, a probabilistic failure of the component i; R is the ultimate loading damage factor D.F. was proposed in Eq. (12) by 0 capacity of the intact structure; R is the ultimate loading Frangopol and Curley (1987), where S and S i intact damaged capacity of the damaged structure. are the mean of the strength of the intact and damaged The lower the loading capacity of the damaged structure, structure, respectively. This damage factor represents the the larger the coefficient of importance and the higher the percent reduction in the mean strength of a given member. sensitive of the structure to the local failure. The member S –S having such large importance coefficient or member D.F.= ------intact damaged (12) sensitivity is called a key member. The importance S intact coefficient γi represents the ratio of the lost of loading capacity due to local failure to the initial capacity of an The above factors are meaures of overall system strength intact structure. Choi and Chang (2009) proposed the idea for intact (R ) and damaged structures (R and R ). A 1 2 3 of a safety ratio ϕ to serve as a acceptance criterion that generalized redundancy was proposed by Pandey and collapse is prevented when λi / λ ≥ 1/ϕ. Based on γi, Barai (1997) as expressed in Eq. (13) assuming that the 0 Huang and Li (2012) proposed a robustness index I as structural redundancy was inversely proportional to the rob the product of reserve strength and residual strength, as response sensitivity. This redundancy measure is quite expressed in Eq (15). general and is applicable to a discrete or a continuum structure. The response sensitivity of each element can be I = min {}k()1–γ (15) computed using the finite element method. However, the rob i i effectiveness of this method depends on the sensitivity Where k is the ratio of ultimate capacity R to the defined in various structural responses of displacement, 0 applied load P. The robustness index is equivalent to the strain, stress, frequency, etc., leading to different degrees ratio of ultimate capacity of damaged structure to the of redundancies. This method can be extended to applied load. If I ≥ 1, the robustness requirement is probabilistic domain if the probabilistic sensitivity is rob satisfied. available. Based on the performance objective of maintaining V sufficient system structural resistance under an extreme ne -----i ∑ S load, Maes et al. (2006) proposed a measure of robustness GR ==------i=1 ij , j 12,,… , j V damage parameters(13-1) based on reserve strength ratio (RSR) defined as the ratio 136 Jian Jiang et al. | International Journal of High-Rise Buildings of the collapse load to the design load of a structure. This elastic-brittle behavior of structural elements, the point of measure has the advantage that it can be increased only first significant yielding in a structural system can be by a structural optimization aimed at maximizing damage considered as a reasonable approximation of the strength S tolerance, and not simply by enlarging the cross section of the nonredundant structure. Therefore, nr in Eq. (17- S of the elements. 1) can be substituted by y , which is the strength of the structural system at the point of the first “yielding”. In RSR RSR based on member i impaired Eq. (17-2), to evaluate the yield strength and the ultimate R ==min ------i ------1 iRSR RSR based on no impaired members strength of a structural system, the average strengths of 0 individual elements were considered during the pushover S S (16) analysis. Then both u and y can be easily identified on the load-deflection curve resulting from the nonlinear Where RSRi is for the damaged structure due to the pushover analysis. failure of the member i; RSR0 is for the intact (undamaged) Izzuddin et al. (2008) proposed a single measure of structure. The measure R1 is taken as the minimum of the robustness in terms of system pseudo-static capacity (Pf) ratios. defined as the maximum bearing capacity by comparing The above robustness measure R1 was used by Masoero the maximum dynamic displacement ud to ductility limit. et al. (2013) to quantify the collapse resistance of structures This measure provides a practical means to assess under bending or pancake collapse mechanism. A “mechanism structural robustness by accounting for dynamic effect, parameter” was defined to indicate the collapse initiation ductility, redundancy and energy absorption capacity. mechanism, based on the mechanical and geometric This approach was also used by Vlassis et al. (2009) for properties of a structure. It was found that R1 was mini- progressive collapse assessment of multi-storey buildings mum for structures under bending collapse both before subject to impact from an above failed floor. and after damage, while R1 was maximum in the case of global pancake collapse. 1 ud Husain and Tsopelas (2004) proposed two redundancy p = max ------P du (18) f ud u ∫ nS d 0 indices, the deterministic redundancy-strength index rs in Eq. (17-1) and probabilistic redundancy-variation index rv in Eq. (17-2). The two indices were to measure the effects of Where Pns is the applied load in nonlinear static element strength on the structural system strength (i.e. analysis. overall effect of redundancy). The former was defined as Tsai (2012) proposed an increase factor for collapse the ratio of mean ultimate to mean yield strength of a resistance defined as the ratio of brace-contributed structure, and the latter was a function of the number of resistance to that of unbraced frame. This factor can be plastic hinges n and their average correlation coefficient related to ductility demand, and can be used to determine ρ between their strengths. The two redundancy indices the design strength and stiffness of added braces to can be calculated for a specific structure and a particular enhance the robustness of retrofiteed buildings. loading condition by performing a static nonlinear analysis Chen et al. (2016b) developed three vulnerability on the structure which is the limitation of this study that indices for robustness of structures under a single discrete dynamic effects were not accounted for. The index rv is event (e.g. only impact), under multiple discrete events probabilistic because it is defined as the ratio of (e.g. earthquake and impact), under continuous events coefficient of variation of the system strength to the (e.g. event during lifetime). In the first case, the importance member strength. coefficient of components γki, as shown in Eq.(14) re- presenting the relation between component behavior and S S system behavior, was used as a weighting coefficient for r ==------u ----u- (17-1) s S S vulnerability coefficient of components (representing the nr y component behavior alone). These indices depended on the damage of components and its effect on the structure, 1+()n–1 ρ and number of possible local damage scenarios. r = ------(17-2) v n 1 S RI = 1–------n γ ⋅v Where u is the ultimate strength or maximum i ∑ ki ki (single discrete event) (19-1) S C1 k=1 resistance of the structure; nr is the strength of the same n structure system as if it were nonredundant by assuming that it consists of ideal elastic-brittle elements. In such a 1 nonredundant structure, the first yielding will lead to RI= ω ⎛⎞1–--- n γ ⋅v ∑ i⎝⎠n∑ ki ki collapse if the strength reserves of the undamaged i k=1 elements have been exhausted. Accordingly, assuming (multiple discrete event) (19-2) Review on Quantitative Measures of Robustness for Building Structures Against Disproportionate Collapse 137

Yan et al. (2019) proposed an index to identify the critical +∞ ⎛⎞1 RI= ω()x ⋅ 1–--- n γ ⋅v member in single-layer latticed domes. This was based on ∫–∞ ⎝⎠n∑ kx kx k=1 the governing collapse mechanism of nodal snap-through (continuous event) (19-3) buckling at either end of the initially removed member. The index estimated the relative vulnerability to node Where RIi is the robustness index of structures under 1 buckling by comparing all indices of all members, without the event i; C is the number of all possibilities when one n explicitly calculating the exact node-buckling load. The out of n components is removed, and its reciprocal, i.e., 1 index was derived by considering influence of the load on 1/C , is a factor to control RI to take the value between n i the node, the stiffness of the connecting members, the 0 and 1. Variables γ and ν denote the importance ki ki boundary condition of the connecting members, and the coefficient and vulnerability coefficient of component k angle of the gap created by the member removal. under event i, respectively; ω is the probability of occurrence i The bearing capacity or collapse mode of an intact or of the event i; ω(x) is the occurrence probability density damaged structure is always determined by finite element function; n is the number of components; analysis. Pantidis and Gerasimidis (2017) proposed analytical Li et al. (2018b) proposed a bearing capacity-based closed-form solutions of collapse load for yielding-type robustness measure in Eq. (20) for steel frames, by collapse mode (due to failure of beams above the removal comparing the load imposed on the intact structure and column) and stability collapse mode (due to buckling of the residual loading capacity of the damaged structure. adjacent columns), respectively. A collapse limit state This robustness index accounted for the dynamic effects factor was defined in Eq. (21) as the ratio of the two and plastic internal force redistribution. I ≤ 0 represents rob collapse loads to determine which collapse mode was occurrence of collapse since the critical load of the triggered first. This factor serves as a indicator of the damaged structure is smaller than the load imposed on the collapse mode: R(a) > 1 applies for yielding-type collapse intact structure, while I > 0 means no collaspe. A rob modes and R(a) < 1 applies for stability collapse modes. hypothetical upper limit for I is equal to 1, and between rob A unity value means simultaneous occurrence of the two 0 and 1, the larger the I value, the better the robustness rob collapse modes. of the frame. C ()a ()q –γq ()q –q Ra()= ------c - (21) I ==------2m 1m------2dm 1m (20) C ()a rob q q b 2m 2dm Where C (a) and C (a) is the collapse load for the Where q is the load on the intact structure, which can c b 1m stability and yielding-type collapse mode, respectively; a be represented by the load imposed on the two middle is the location of the column removal. bays adjacent to the removed column, while load imposed on the side bays is denoted as q as shown in Figure 6a. 1s 5.1.2. Deformation-based robustness index The maximum values of q and q can be considered as 1m 1s Biondini et al. (2008) presented a robustness index the elastic limit load of intact structure; q is the static 2m associated with the displacements of the system as shown critical load of the middle bays for the damaged structure in Eq. (22). This index is related to the load ratio and until failure (Figure 6b) and q is the corresponding static 2s damage degree of a structure. critical load of the side bays for the damaged structure; γ is a dynamic amplification factor, and q2dm is the dynamic s ρ = ------0 critical load of the middle bays for the damaged structure s (22) d until failure, which may be obtained by q2dm = q2m/γ. Different from most above methods for frame structures,

Figure 6. Schematic of structural load on a structure: (a) intact structure; (b) damaged structure. 138 Jian Jiang et al. | International Journal of High-Rise Buildings

Where s is the displacement vector, . denotes the generality, can be met by these damage-based approaches. euclidean scalar norm, and the subscripts “0” and “d” The question of calculability, on the other hand, is more refer to the intact and damaged state of the structure, difficult to answer. This is because determination of the respectively. total damage resulting from initial damage requires an Huo et al. (2012) proposed a ductility coefficient of examination of the failure progression which is always rotation capacity of beam-to-column connections to achieved by dynamic analysis, taking into account geometric measure the robustness of steel structures, defined as the and material non-linearity, as well as separation and ratio of ultimate rotation capacity θmax at failure to the falling down of structural components. Depending on the rotation θc at the formation of catenary action in beam, as type of structures and the governing type of collapse, the given in Eq. (23). When ϕj > 1, it suggests that the damage-based approaches will become too complex and connecton is able to develop catenary in the beam. The intractable for practical design purposes. higher the ductility coefficient, the higher the reserve capacity of the connection and the structure. 5.1.4. Energy-based robustness index The above discussion shows that stiffness-based θ ϕ = ------max- (Section 4) and damage-based measures of robustness are j θ (23) c either easy to calculate or expressive, but not both. Approaches based on energy considerations might be 5.1.3. Damage extent-based robustness index capable of meeting both requirements to the same degree. Based on the quantification of the damage progression Energy-based methods are based on the assumption that resulting from initial damage, a dimensionless damage- total external work done by the applied load energy based measure Rd of robustness was proposed by Starossek flowing into a system must equal the total amount of and Haberland (2011) as: energy in the system, as a sum of internal energy (strain related energy) and kinetic energy (velocity related energy). p R = 1–------If a collapsing structure is capable of attaining a stable d p (24) lim energy state through absorption of gravitational energy, then collapse will be arrested. Otherwise, if a deficit in where p is the maximum total damage resulting from energy dissipation develops, the unabsorbed portion of the initial damage; plim is the acceptable total damage. released gravitational energy is converted into kinetic Note that p and plim refer to damage in addition to the energy and collapse propagates from unstable state to initial damage. The quantification of damage required unstable state until total failure occurs. herein can be performed with regard to the affected Zhang and Liu (2007) proposed a network of energy masses, volumes, floor areas or even the resulting costs. flow for framed structures. An energy-base member i Another formulation Rd,int of a damage-based measure importance coefficient γ was defined in Eq. (26) as the of robustness was proposed by Starossek and Haberland ratio of energy stored in the damaged structure U(i) after (2011) as the complement of the integral of the dimensionless member removal to the total energy U of the intact damage progression caused by various extents of initial structure. The more the deformation energy in the damaged damage i. A value of one stands for maximum possible structure, the more importance the removed member. The robustness and a value of zero stands for a total lack of network can be used to find the key members and also the robustness. In Eq. (25), initial damage of any size is critical load redistribution path. However, it is more considered. However, an initial damage larger than a limit applicable to be a qualitative indicator rather than value specified by design objectives can no longer be quantitative measure of structural robustness since the considered as local, which is one limitation of this coefficient varies in a range of (1, +∞) and its relation to method. the member importance is nonlinear.

R = 12– 1 []di()–i di U()i ∫ (25) γi = ------0 U (26) Where d(i) is the maximum total damage resulting from the initial damage of extent i. Both d(i) and i are Huang and Wang (2012) proposed an importance index dimensionless variables obtained by dividing the respective CIi of structural members in Eq. (27-1) based on their reference value (mass, volume, floor area or cost) by the contributions to the structural energy distribution and the corresponding value of the intact structure. damage-affected areas. The larger the coefficient, the In addition, the measure Rd in Eq. (24) is expressive more important the member and the more critical the since it directly refer to the design objectives of “acceptable initial damage scenario. They insisted that redundancy total damage”, while the integral measure Rd,int in Eq. (25) depended on the response sensitivity and the reserve is thus not expressive. The requirements of expressiveness strength of the members (safety margin). A redundancy and simplicity, and in principle also the requirement of index Rj was also proposed in Eq. (27-3) based on the Review on Quantitative Measures of Robustness for Building Structures Against Disproportionate Collapse 139 strain energy-based sensitivity of individual members and E R = 1–max------rj,- the loading capacity redundancy. The measure of robustness e E (30) was taken as the minimum of all the redundancy indices j fk, for different member removal scenarios. However, this method is applicable to removal of one member and does Where Er,j is the energy released during the initial not consider the failure of connections. failure of a structural element j and contributing to damaging a subsequently affected structural element k; α S CI = ------i -⋅------i Ef,k is the energy required for the failure of the sub- i max{}α max{} S (27-1) i i sequently affected structural element k. The Eq. (30) has a simple form and also appears to be expressive with regards to the possibility of a total S = ()Nn–1+ ⋅S (27-2) i i0 collapse. A difficulty, however, arises in determining the value of Er,j, which can be overderestimated or underesti- ne β mated (Haberland 2007). The energy released by the ∑ ()≠ ij, R = ------i=1 ij - (27-3) initial failure of the structural element consists of several j ne ()β , /c , ∑i=1()ij≠ ij ij parts. In structures vulnerable to pancake-type or domino- type collapse, the gravitational potential energy of sepa- th Where CIi is the importance index of the i member; αi rating, overturning or impact elements is transformed into is the variation of the total strain energy of the structure kinetic energy in a discrete and concentrated manner, before and after the damage of the ith member; N and n which makes a major and even dominant contribution to is the total number of storeys and the storey of the ith the total released energy. In such cases, a numerical assess- member, respectively; Si0 is the affected area due the ment can be easy and effective. However, in structures th damage of the i member; Rj is the redundancy index for susceptible to other types of collapse (e.g. yielding-type th the damage of the j member; βij is strain energy collapse), the value of Er,j can only be determined through th sensibility defined as the change of strain energy in the i a complete structural analysis. Furthermore, Er,j should th member due to the damage of the j member; ci,j is the include only the energy portion that contributes to damaging capacity redundancy factor taken as the minimum of the the subsequently affected element k. This estimation reserve strength-based value and stability-based value. again is relatively easy in structures having pancake-type Fang and Li (2007) proposed an energy-based robustness or domino-type collapse modes. Therefore, it is expected index Irob1 by comparing the total internal energy Eu of an that such structures are the most suitable ones for the intact structure at the ultimate limit state and that Ed at the application of energy-based measures of robustness. design limit state (e.g. yielding), expressed as: Bao et al. (2017) conducted nonlinear static pushdown analysis to determine the ultimate loading capacity of a E I = -----u damaged structure given a column loss scenario, and rob1 E (28) d applied an energy-based analysis to account for the dynamic effect due to sudden column loss. The dynamic This index can only represent the performance of the effect was considered by assuming that the external work intact structure, rather than the robustness of the structure was totally transformed into internal work at the peak in the case of local damage. To this end, this index was dynamic displacement where the kinetic energy is zero, improved by Lv et al. (2011) to consider the internal and the static load-displacement curve from pushdown energy Er of the damaged structure at its ultimate limit analysis was thus modified to obtain a dynamic load- state: dynamic peak displacement curve. The dynamic ultimate capacity of the damaged structure, representing the peak E E E I ===-----r, I -----r, I ------u - vertical load that can be sustained under sudden column rob2 E rob3 E rob4 E –E (29) d u u r loss, was determined corresponding to the peak static load. A robustness index R was proposed in Eq. (31) as By comparing the energy released during an initial the minimum value of the normalized ultimate capacity λi failure and the energy required for progression of failure, du, over all damage scenarios, which is similar to the Starossek and Haberland (2011) proposed an energy- concept R2 in Eq. (9-2). A unity value of R indicates no based measure Re of robustness. This simple formulation collapse for all sudden column loss scenarios. This is to assess the possibility of a total collapse with a value method avoids the requirement of complex dynamic of one indicating perfect robustness. Values between one analysis, and enables the use of various dynamic increase and zero are acceptable to a greater or lesser degree, factors in the same structure (different dynamic increase while negative values indicate failure progression to factors used for different column removal locations). The complete collapse. proposed index is more readily calculable than the damage- based approaches in Section 5.1.3 that require calculating 140 Jian Jiang et al. | International Journal of High-Rise Buildings the extent of damage. This is because the analysis can be measures, i.e. failure probability-based, risk-based and terminated at the ultimate load for determining R and it is reliability-based measures, respectively. The first category not necessary to analyze the spread of damage following considers the uncertainties in abnormal events and structural the formation of a collapse mechanism. However, this is system, while the second involves the probability of the also the limitation of this method. The energy-based causes (abnormal events) and consequences of structural procedure was based on the assumption of an unchanging collapse. The third one, as a trade-off, accounts for radomness deformation mode for both static loading and sudden in structural system alone. column loss, which may give nonconservative results in P = Φ()–β the post-ultimate response when failures caused the f (32) change of collapse modes. Where Φ is the cumulative Gaussian probability Rmin= ()λi i∈ column removal scenarios i du, (31) distribution function.

Beeby (1999) proposed using of energy dissipation per 5.2.1. Failure Probability-based Robustness Index unit volume of the intact structure as a measure of Failure probability-based robustness indices take into robustness. Smith (2006) defined a measure of robustness account probability of abnormal events and local failure as the minimum energy required to destroy enough members as well as probability of collapse given the local failure. to cause the collapse of a structure. Different structural They are always derived by comparing the probabilities arrangements can be compared in this way and critical of the system failure for an undamaged and a damaged sequence of collapse can be identified. The buckling structure. energy and failure energy of columns were distinguished Ellingwood (2005, 2006) proposed that the probability by Szyniszewski (2009) and Szyniszewski and Krauthammer of building collapse must be limited to a socially accepted (2012), considering that the column buckling did not value (e.g. 10-6/year). Let H be the abnormal event and D always lead to column failure and collapse propagation. be local damage. The probability of structural collapse An energy-based demand capacity ratio was proposed due to H is expressed as (Ellingwood 2005; Starossek and based on the post-buckling deformation energy of damaged Haberland 2008): columns. A comparison to force-based demand capacity ratio showed that the force-based ratio was not very sensitive P[]Collapse = P[]Collapse D]PDH[]P[H (33) to the fundamental changes in structural behavior, and the column deformation energy was a better stability indicator Where P[H] is probability of H; P[D|H] is conditional under dynamic loading than the maximum dynamic force. probability of damage state D, given H; P[Collapse|D] is Recently, Wilkes and Krauthammer (2019) investigated probability of collapse, given damage state D. the correlation of energy flow and rate of energy flow to For multiple hazards and damage states, Eq. (33) can member failure and structural collapse of mid-rise steel be rewriten as: framed structures. This was achieved from energy-time curves obtained by nonlinear dynamic analysis. P[]Collapse = P[]Collapse[]D PDH[]P[H ∑H∑D (34) 5.2. Probabilistic Robustness Index The deterministic robustness or redundancy indices are The probabilities P[D|H] and P[Collapse|D], rather a fixed value used to identify critical members and to than controlling abnormal events P[H], are within the assess performance of a structural system for given damage control of a structural engineer. The common design scenarios. It is difficult for them to include all the possible standards usually describe safety of a structure as a damage scenarios and potential failure paths. To account function of safety of all elements against local failure, and for the random nature of the required information such as therefore only take into account the probability P[D|H]. uncertainties in abnormal loads, member and system The reaction of a structure to the possible occurrence of responses, consequences of failure, probabilistic measures local failures in terms of P[Collapse|D] is seldom investi- should be used to quantify robustness of structures, from gated. Only important buildings are investigated regularly a more practical view. A common probabilistic measure for this case. A summary of the probability of terrorist of safety is the reliability index β which is related to the threat, hazard, damage, fatality, and economic and social probability of failure Pf through Eq. (32). The probability loss for progressive collapse is described by Stewart of failure Pf can be used to directly measure the robustness (2017) and Adam et al. (2018). of structures. When it is difficult or impossible to To overcome the limitation of deterministic measures calculate the failure probability, the reliability index β can that only the ultimate limit state is considered, Lind (1995) be used to indirectly measure the failure probability and proposed complementary concepts of damage tolerance structural robustness. The following subsections present a in Eq. (35-1) and vulnerability in Eq. (35-2) to capture the review on three main categories of probabilistic robustness reduction in reliability of a system that was damaged Review on Quantitative Measures of Robustness for Building Structures Against Disproportionate Collapse 141

without collapse. The damage tolerance Td is a function i. This measure is related to the criticality of individual of a set of system states (different levels of damage) and members and their role in a redundant system, which can loading conditions. For a specific damage state and loading be related to a variety of importance-based measures condition, a damage factor was defined as the ratio of commonly used in system reliability. failure probability of a damaged state to an undamaged P state. The vulnerability V was defined as the reciprocal of R = min ------S0 2 i P (38) the damage tolerance. The proposed measure is a step Si toward differentiation of a set of possible system states rather than simply considering “failure” and “no failure” Felipe et al. (2018) proposed a systematic reliability- in normal probabilistic analysis. based approach to identify key elements in a redundant structure (the element most likely to cause collapse) and T = PR(),S /PR(),S d 0 d (35-1) to simplify the design process. The approach involved a threat-specific analysis, in which probabilities of initial damage were evaluated for a given loading event. Damage VPR= (),S /PR(),S d 0 (35-2) propagation and ultimate collapse were distinguished to rank elements in a system in terms of their vulnerability Where Td is the damage tolerance; V is the vulnerability; regarding structural collapse. The key element was P() is the probability; R0 and Rd is a set of undamaged and identified as the one presenting the largest intersection damaged states, respectively; S is a set of prospective between vulnerability and importance with respect to loadings. collapse. A coefficient of vulnerability (CV) of an element Frangopol and Curley (1987) and Fu and Frangopol was defined as the ratio of failure probability of the (1990) proposed a probabilistic measure related to element P[fi] to the sum of failure probabilities for all structural redundancy (RI), which also indicated the level elements in the system. A coefficient of importance (or of robustness: disproportionate) CID for damage propagation of an element was defined as the ratio of the probability of P –P []f f RI= ------f() damaged f() intact- failure progression P j i to the maximum possible P (36) f() intact probability of damage propagation. A coefficient of im- portance CIC for progressive collapse of an element was Where Pf (damaged) and Pf (intact) is the failure probability of defined as the ratio of the probability of occurrence of the th a damaged and intact structural system, respectively; This failure path ci, starting with the failure of the i element redundancy index provides a measure of redundancy of a to the maximum probability of failure path. A coefficient structural system, ranging between zero and infinity with of vulnerability CVD to damage progression of an a lower value indicating a higher robustness. However, it element was defined by the product of CV and CID. A could be difficult to assess this index in a practical appli- coefficient of vulnerability CVC to structural collapse of cation due to the wide range of the values that the index an element was defined as the product of CV and CIC. can take. The values of the above coefficients are always between By replacing Pf (intact) in Eq. (36) with acceptable failure 0 and 1 from their definitions. The key element of a probability Pf (acceptable), a new robustness index was system was defined as the element having the largest proposed by Chen et al. (2016a), as expressed in Eq. (37) vulnerability with respect to structural collapse. Design where Pf (collapse) = P[Collapse]. Based on the robustness improvements toward structural integrity should focus on index, the performance of a structure can be classified the key element. into four levels: collapse (RI ≤ 0), low robustness (0 < However, the proposed methodology is a threat- RI ≤ 1/3), general robustness (1/3 < RI ≤ 2/3), high robustness dependent method (depend on failure probabilities of the (2/3 < RI ≤ 1). This classification is important for design structural elements), which is rather complex in the sense purpose. that it requires full probabilistic analysis of all possible failure paths and identifies element vulnerability and the P –P RI= ------f() damaged f() intact- key element. This may render the applicability to large P (37) f() intact structural systems limited.

Based on the performance objective of maintaining Pf[] CV = ------i - sufficient system reliability under an extreme load, Maes i (39-1) n Pf[] et al. (2006) proposed a measure of robustness R2 expressed ∑ j=1 j as the minimum of system failure probability ratio for each member. The ratio was defined by comparing the P []f p system failure probability P of the intact structure to that CID = ------i - (39-2) s0 i max []P []f p m m Psi of the damaged structure due to one impaired member 142 Jian Jiang et al. | International Journal of High-Rise Buildings

PC[] P –P CIC = ------i R ()kl, = ------kl, k,()1≤≤lm i max []Pc[] (39-3) I P (40-2) m m k

KE== max []CVC max []CV ⋅CIC i i i i i (39-4) p R ()l ===s ------k -,(k 12,,… ,s) P ∑ η R ()kl, ,η k=1 k I k s p Where ∑k=1 i (40-3) P []f p ==∑n P[] f f ⋅P[] f and m 1,,… n m j=1,i≠–1 j i i p (39-5) R ==s m ------kl, , w ∑ η ∑ ω R ()kl, ,ω , k=1 k l=1 kl, I kl m p ∑j=1 kj,

PC[]= n P[] f ⋅⋅P[] f f P[]()U () f f ()k=12,,… ,s;l=12,,…m (40-4) i ∑j=1,ji≠ i j i kj, ≠ ik, ≠ j k i ,j

a() j (39-6) Where Ri is the strength of the component ri; i is the internal stress of component ri when imposing an unit Lin et al. (2019) proposed a novel methodology to load on the damaged structure consisting of rj, rj+1, …, th quantitatively evaluate the structural robustness of rn; Pk is the occurrence probability of the k failure path; th offshore platforms against progressive collapse. A Pk,l is the structural failure probability at the l failure λ()j th generalized bearing capacity ratio ( i ) was defined as step of the k failure path; n is the number of com- the change of internal stress in the ith component before ponents; m is the number of failure steps for a failure and after failure of the jth component. The component path; s is the number of failure paths. with the maximum ratio was regarded as the candidate failure component at the jth failure step of the mth failure 5.2.2. Risk-based Robustness Ratio path, and thus the total failure paths and all failure steps Risk-based robustness assessment offers a powerful in each path were obtained. An incremental loading and full probabilistic framework, by considering the method was used to calculate the reliability of each probability of the causes (i.e. how likely is the hazard failure path, based on which the probability of each event) and consequences (i.e. what happens when such failure path was determined. Three robustness indices events do occur such as loss of safety and economy). were proposed: path and state-dependent robustness However, complexity and subjectivity reduce the calcula- index RI, overall robustness index RP, and comprehensive bility and application potential of the risk-based approaches. robustness index RW. The RI describes the robustness of Without some appreciation for these risks, it is difficult to the structure at each step of each failure path, and can judge the effectiveness of various strategies to mitigate illustrate the effect of each local damage on the failure structural collapse. probability of a structure. The overall robustness index RP Based on the definition of risk, Pinto et al. (2002) is expressed as a weighted sum of RI using a path weight defined the risk SR of a failure as the product of the ηk, demonstrating the overall effect of various potential probability Pf of its occurrence and its consequence C, as failure paths on structural robustness. Performing twice given in Eq. (41). The consequence C can be defined in weighting on RI by path weight ηk and step weight ωk,l the context of ultimate limit state or servicebility limit leads to the comprehensive robustness index RW, which state. includes both path information and state information at the same time. It was found that the robustness varied SR = Pf × C (41) significantly under different failure paths, and an unexpected accident occurring in a small-probability path would have Risk-based quantifications of system robustness were a more serious impact on structural robustness. One problem first proposed by Baker et al. (2008). A robustness index of this study is that the deduction of probability of failure in Eq. (42) based on the definition that “a robust system paths is not clear, and it seems that no dynamic effect was is considered to be one where indirect risks do not con- included in the analysis. tribute significantly to the total system risk” (Formisano et al. 2015). The approach divided consequences into

() direct consequences associated with the damage of a local ⎧⎫a j λ()j ==------i -, j 1 element and indirect consequences associated with the ⎪⎪i R ⎪⎪i additional and subsequent system failure. The robustness ⎨⎬, (i=1,2,…,n) (40-1) measure resulted from the comparison of direct risks with ()j ()j–1 ()j ⎪⎪() ⎛⎞a a a the total risks (sum of the direct and indirect risks). The ⎪⎪λ j = ⎜⎟------i -⁄------i =------i ,()j>1 i R R ()j–1 ⎩⎭⎝⎠i i a system risk was computed by multiplying the consequence i of each scenario by its probability of occurrence, and then Review on Quantitative Measures of Robustness for Building Structures Against Disproportionate Collapse 143 integrating over all of the random variables. The fewer the exposures (e.g. human/gross errors), a conditional indirect risks involved in the total risk, the more robust a robustness index was proposed by Sørensen (2011) using structure is. The index takes values between zero and one, risks RDir|exposure and RInd|exposure conditioned for a given where IRob = 1 represents a completely robust structure exposure as: since there is no risk due to indirect consequences, while R IRob = 0 denotes a completely vunerable structure that all I = ------Dir exposure - rod exposure R +R (44) risk is due to indirect consequences. This measure of Dir exposure Ind exposure robustness allows the robustness of different systems to be compared. A vulneribility index Iv was also proposed by Baker et However, one problem of this method is that it al. (2008) as the ratio of total direct risks to the total measures the relative direct risk due to indirect risk, direct consequences. This vulnerability index provides an resulting in a false impression that a system might be indicator of the risks associated with structural damage, deemed robust if its direct risk is extremely large (relative normalized by the direct risk exposure. to its indirect risk), but that system should be rejected based on reliability criteria (i.e. predefined acceptable R ∑i Dir direct risk) rather than robustness criteria. Sørensen I = ------i- (45) V C (2011) argued that the robustness index in Eq. (42) was ∑j Direlj not always fully consistent with a full risk analysis, R although it was a helpful indicator based on risk analysis Where Diri is the direct risk due to the ith damage; C th principle. The direct risks can be estimated with higher Direlj is the direct consequence due to the j damage. accuracy than the indirect risks, since the direct risk Maes et al. (2006) proposed a risk-based robustness typically are related to code-based limit states. measure R3 that served as an indicator for the per- formance objective of containing the costs associated R I = ------Dir - with the consequences of failure. It was quantified as the Rob R +R (42) Dir Ing inverse of the tail heaviness H of the log-exceedance curve based on the relation of failure consequences Where IRob is the robustness index; RDir is the direct versus probability of exceedance. The tail heaviness is a risk; RInd is the indirect risk. frequently used quantitative measure of risk, and can be Faber et al. (2017) revised Eq. (42) and proposed a easily computed numerically for a specific consequence- more general and consistent scenario-based approach to logexceedance probability curve (Maes 1995). H < 1 re- quantify robustness. The revised formulation took the presents robustness (i.e. a fully contained consequences), ratio between direct consequences and total consequences while H > 1 means non-robustness (i.e. out-of-control as scenario wise, and the robustness index with respect to consequences). Note that the measure R3 is appropriate to a given scenario I is expressed as in Eq. (43). The evaluate the robustness of a system subjected to an selection of the formulations depended on the focus of the exceptional hazard, since it considers both the con- system assessment. Note that the robustness index IR(i) sequences of failure including follow up consequences itself is a random variable which may be analysed further and their likelihood. In practical situations, it is usually by categorization and ordering of the different scenarios both appropriate and efficient to consider “easy” measures in accordance with the hazard, damage, failure and of robustness such as R1 in Eq. (16) and R2 in Eq. (38). consequences. A more general method based on a full consequence analysis is valuable if one wishes to give a quantitative c ()i I ()i = ------D meaning to “robustness” in the case of more complex or R c ()i (43-1) T continuous human, environmental and engineered systems. R = 1/H c ()i 3 (46) I ()i = ------DI, - R c ()i +c ()i (43-2) DI, DP, 5.2.3. Reliability-based robustness index As a trade-off, reliability-based robustness assessment c ()i +c ()i I ()i = ------DI, DP, approaches only account for the randomness of structural R c ()i ++c ()i c ()i (43-3) DI, DP, ID systems by ignoring the probabilities of the accidental events and the corresponding initial damages. Reliability- Where CD(i) and CT(i) is the direct and total con- based measures focus more on progressive collapse sequences, respectively; CD,I(i) and CD,P(i) is the direct resistance of a structure itself, and define the robustness consequences due to the initial damage and propagated as a function of the failure probabilities. Once the damage, respectively; CID(i) is the indirect consequences. reliabilities of the intact and damaged structure subjected To avoid the difficulty in quantifying the probability of to initial damages are obtained, the structural robustness 144 Jian Jiang et al. | International Journal of High-Rise Buildings can be computed easily. Gharaibeh et al. (2002) presented a system reliability- If the resistance R and the applied load P follow based methodology to identify and rank important common probability distributions (known probability density members in structural systems under different material function) such as normal or lognormal distributions, the behaviors (brittle or ductile) and for different stiffness reliability index can be mathmatically determined based sharing factors. The importance of a member was defined on the mean value and standard deviation of R and P. For as its impact on the system reliability. Two member random variables R and P with unknown probability importance factors were proposed: member reliability density function, a reliability analysis is needed. The importance factor in Eq. (50-1) and member post-failure reliability index can be evaluated on a member-by- importance factor in Eq. (50-2). The former represents member basis or on a structural system basis. the sensitivity of system reliability βsystem to changes in Based on the system reliability, a probabilistic redun- reliability of the member βm,i (with a normalized form of β I0 dancy index R was proposed (Frangopol and Curley mi, ), while the latter measures the sensitivity of system 1987; Fu and Frangopol 1990; Sørensen 2011) as: reliability to changes in member post-failure behavior (residual strength). The residual strength of the damaged β η β = ------intact member was measured by a strength factor i depending R β –β (47) η intact damaged on the ductility of the member ( i = 0 for perfect brittle behavior; ηi = 1 for perfect ductile behavior). A simple Where β intact and β damaged is the reliability index for the expression of reduction in reliability index of the structure collapse limit state of the intact and damaged system, given a member failure was also proposed. Although respectively. This index takes values between zero and these factors can be used for complex structures, they infinity, with larger values indicating larger robustness. needs reliability analysis based on statistic data and Ghosn and Moses (1998) measured the robustness of probability distrubiton of load and capacity, and are bridge structures in terms of the reliability index margin difficult to be used in practical engineering. between system reliability and the reliability of the key ∂β member. system 0 N I , = ------,I , =I , / I mi ∂β mi mi ∑j=1 mj, mi, Δβ = β –β u intact member (48-1) (reliability importance factor ) (50-1) Δβ = β –β I ==β |η 1.0–0.0β |η = f functionality member (48-2) η,i system i system i Δβ = β –β (post-failure importance factor) (50-2) d damaged member (48-3) The deterministic redundancy-strength index r in Eq. Where β is the reliability index for the fun- s functionality (17-1) and probabilistic redundancy-variation index r in ctionality (servicebility) limit state of the intact structure; v Eq. (17-2) proposed by Husain and Tsopelas (2004) β is the reliability index of the most critical member. member correspond to the mean value and coefficient of variation, An equivalent reliability-based robustness index ranging respectively, in the statistic theory. These two indices between zero and unity was proposed by Sørensen (2011) were further used to evaluate the redundancy response as: modification factor and reliability index by Tsopelas and β Husain (2004). The modification factor was used to I = ------damaged rob β (49) modify the static analysis results by considering the effect intact of redundancy. The reliability index β is expressed as Eq. (51). This index is a function of the system redundancy Feng et al. (2020) applied probability density evolution (r , r ), coefficient of variation (COV) of the strength v of method (PDEM) to determine the system reliability of s v the elements in the structure, COV of the load l on the reinforced concrete structures subjected to progressive structure. This index represents a good step to measure collapse. The structural uncertainties including geometric the relation of reliability and redundancy. Further work is properties, material properties and applied loads were needed to extend it using nonlinear dynamic analysis. considered. The PDEM was employed to calculate the probability density function of the collapse resistances for ()r –l these uncertainties obtained from pushdown analysis. The β = ------s - (51) v ⋅ r2⋅r2+l2⋅v2 critical damage scenario together with PDEM was used to e v s determine the failure reliability. The robustness index in Eq. (47) was used to measure the robustness of structures. Liao et al. (2007) proposed a uniform-risk redundancy One limitation is that independent random variables were factor to modify the design lateral load for steel moment used which cannot reflect interaction between coupled frames under earthquake. The redundancy factor RR was uncertainties. defined as the ratio of elastic spectral acceleration at the Review on Quantitative Measures of Robustness for Building Structures Against Disproportionate Collapse 145 fundamental period causing collapse at the two pro- The former is based on the structural topology (con- bability levels (Pic is the actual probability of collapse; figuration) and structural stiffness (with or without Pall ic is the allowable probability of collapse). Although load), while the latter depends on the structural this method is seismic-purpose, it can be potentially performance reponses such as bearing capacity, defor- applied to progressive collapse by replacing the spectral mation, damage extent, energy flow etc. Based on acceleration with dyamic increment factor. whether considering the probability concept, the structural performance-based measures can be further ⎛ classified into deterministic and probabilistic measures. ⎜ 3. The deterministic measures predominate in quanti- R = ⎜ (52) R ⎜ fying structural robustness, which are always derived ⎝ by comparing the structural responses of an intact and damaged structure. They are mostly expressed by The largest challenge in reliability-based robustness dimensionless factors in terms of bearing capacity, quantifications is the calculation of system reliability. The displacement or rotation, damage extent or energy. most widely adopted method is the Monte Carlo simulation, They are largely characterized by a systematic analysis which is efficient but suffers from a large computational for a range of column removal scenarios (alternate cost. An alternative method is to use the probability load path method) using nonlinear static and dynamic density evolution method (PDEM) (Feng et al. 2020), analysis or energy-based method. which is derived based on the probability conservation 4. The structural attribute-based measures of robustness principle. The PDEM establishes a new method to reach are derived from algebraic properties (determinant, the balance between computational efficiency and accuracy condition number, matrix norm) of system stiffness in system reliability assessment. matrices. Some of them lack clear physical meanings, which can only be used as indicators of robustness in 6. Summary and Recommendations a qualitative manner. Most attribute-based measures are applicable to the elastic state of simple structural 6.1. Summary of Existing Quantification Approaches form (e.g. truss structures) and their extention to other Insensitivity to local failure is referred to as robustness. types of structures (e.g. frame structures) and plastic Different structural systems subjected to different local state needs further work. failures may suffer from different collapse modes, and 5. The probabilistic measures are receiving growing thus exhibit different degrees of robustness. Such differences interest because it is difficult to consider all fore- are not included in modern probability-based design seeable damage events in deterministic measures. procedures using partial safety factors. Additional con- They involve the probability of uncertainties in abnor- siderations are therefore necessary to ensure structural mal events, structural system and failure-induced robustness after an initial local failure. Such considerations consequences, and can be further classified into have been made in the past mostly in qualitative form failure probability-based measures (include uncertainties rather than quantitative manner. This paper presented a in abnormal events and structural system), risk-based comprehensive review on quantitative measures of structural measures (include probability of the causes and robustness, and the following findings can be drawn: consequences) and reliability-based measures (a trade- 1. A clear definition of robustness is the precondition of off method involving radomness in structural system a better quantitative measure of robustness. Robustness alone). is a property of the structural system alone, i.e. a 6. Nonlinear static analyses dominate the determination system behavior independent of the possible abnormal of peformance of intact and damaged structures, events and probabilities of the induced initial local serving as a trade-off between linear static and nonlinear failure. Robustness can be achieved by continuity (tie dynamic analysis. The dynamic increase factors are strength), redundancy (alternate load path), ductility, always needed to consider the dynamic effects due to segmentation, energy absorption capacity, etc. It is sudden removal of columns. It is also needed to select different from collapse resistance (a broader concept bay pushdown analysis (increasing gravity loads only also dependent on initial damage), vulnerability (system on the damaged bays) or uniform pushdown analysis or component behavior for a given initial failure) and (increasing gravity loads on all bays) to determine redundancy (one aspect of robustness to provide the capacity of structures. multiple alternate load paths). 7. Robustness is still an issue of controversy for its 2. A total of about 50 quantitative measures of quantification. The purposes of existing quantitative structural robustness are found in terms of various measures of robustness differ, focusing on ranking robustness or redundancy indices, which can be structural members, identifying critical components, classified into two main groups: structural attribute- failure paths and collaspe modes in a structural system, based and structural performance-based measures. accounting for aspects of the mebers including location 146 Jian Jiang et al. | International Journal of High-Rise Buildings

in the system, reserve strength, residual strength, indices for these individual hazards, respectively, and stiffness sharing, and material behavior (ductile or choose a optimized structural system by comparing brittle). Other purposes are to check the usefulness of these measures. On the other hand, it means to develop robustness recommendations, indicators and prescri- a robustness measure to quantify robustness of a ptions. There is no generalized form of quantitative structure under a realistic combination of hazards (e.g. measures to comprehensively assess the robustness fire after earthquake, fire after blast). of structures with a single robustness index. 5. The robustness indices always fall in a range between 0 and 1, representing two extreme situation 6.2. Issues and Recommendations of completely robust and completely vurnerable. Most 1. A generalized quantification of robustness needs to measures are not related to performance objectives start from the generally accepted definition and (i.e acceptable structural response) such as acceptable design objectives, and to focus on the response of a capacity limit, deformation limit, extent of collapse. structure to local damage as well as the associated Thus, a further classification of the degree of robustness consequences of collapse. The inclusion of collapse is lacking (e.g. collapse, low robustness, moderate consequences is the requirement to be used as a robustness, high robustness). decision-making tool. A generalized quantification 6. For the design purpose, deterministic measures are can be achieved by comparing a set of deterministic predicated on the assumption that robustness is a and probabilistic robustness indices rather than using variable property of the structure that can be calibrated a single index. to meet a fixed design load via increased strength and 2. Many quantitative measures of robustness are improved load path redundancy. One limitation of applicable to one column removal scenarios, simple deterministic measures is that only the ultimate limit structural types idealized into a set of parallel-series state is considered since robustness changes throughout components (e.g. truss, frame structures), regular structural the entire collapse process (each failure step in each layouts, predefined single collapse modes (yielding- failure path). Another limitation is that they cannot type model or stability mode). Therefore, further cover all the local damage scenarios, and do not research is needed for multiple column removal consider the consequences of collapse at overload, scenarios (realistic damaged components in the blast- which cannot be used for decision-making. Therefore, it affected zones), complex structural types (discrete or is recommended that deterministic measures are used continuum), irregular structural layouts (vertical or to evaluate the robustness of the structural system horizontal), and multiple or coupled collapse modes. alone in the early design stage through alternate path 3. The existing design methods and quantitative measures method. Probabilistic measures, as more reasonable of structural robustness are proposed and applicable and practical tools, can be used for decision-making for steel framed structures and reinforced concrete on the selection of enhancing strategies for robustness structures. These are not directly applicable to large- from the view of safety and economic costs. span structures, prefabricated steel structures, precast/ 7. The energy-based measures of robustness are more prestressed concrete structures, modular struc-tures, applicable to structures susceptible to pancake-type timber structures, offshore platforms, which are or domino-type collapse where the energy flow can inherently susceptible to progressive collapse due to be easily determined since the energy transform from lack of continuity because of discrete and potentially gravitational potential energy to kinetic energy in a brittle connections, large spans, and heavy elements. discrete and concentrated manner. For other types of In addition, these types of structures have different collapse such as yielding-type collapse, the energy collapse modes and resisting mechanisms, and thus transformation can only be determined through a needs different measures to quantify their robustness. complete structural analysis. 4. Many quantitative measures of robustness are appli- 8. The quantitative measures of robustness are always cable to single hazard such as earthquake, blast, derived by conducting nonlinear static or dynamic impact or fire. Progressive collapse design require- analysis, which is complex and time consuming. It is ments can add major cost to building construction if recommended to use machine learning method (i.e. it is achieved without consideration of other design artificial neural network) for fast prediction of the requirements. By taking a multi-hazard approach collapse mode and failure patterns of structures as progressive collapse mitigation can often be achieved well as subsequent progressive collapse potential with minimal additional construction costs. It is assessment. A systematic methodology based on both therefore recommended to extend the existing robustness Monte Carlo simulation and probability density measures for multiple hazards, in an attempt to evolution method should be developed to generate a reduce risk of progressive collapse, improve structural robust and suffificient large dataset for training and life safety and save economic costs. On one hand, testing. this means to develop different forms of robustness 9. Large-scale experiments by loading structures to Review on Quantitative Measures of Robustness for Building Structures Against Disproportionate Collapse 147

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Appendix A. List of definitions of disproportionate and progressive collapse Source Definition of disproportionate or progressive collapse The spread of an initial local failure from element to element, eventually resulting in the col- ASCE 7 2010 lapse of an entire structure or a disproportionately large part of it. Progressive collapse is a situation where local failure of a primary structural component leads GSA 2005 to the collapse of adjoining members which, in turn, leads to additional collapse. Hence, the total damage is disproportionate to the original cause. Progressive collapse. A chain reaction failure of building members to an extent disproportion- UFC 4-010-01 DoD 2013 ate to the original localized damage. Progressive collapse-The spread of local damage, from an initiating event, from element to NISTIR 7396 NIST 2007 element, resulting, eventually, in the collapse of an entire structure or a disproportionately large part of it; also known as disproportionate collapse. The building shall be constructed so that in the event of an accident the building will not suffer Building Regulations 2010 collapse to an extent disproportionate to the cause. In cases when loads are larger than assumed in the design wok, the base metal will not show plastic deformation and will cause collapse. Because peripheral members including connec- JSSC 2005 tions are then subjected to increased loads, the collapse further advances. This chain-reaction fracture and collapse phenomenon are called progressive collapse. Progressive collapse can be defined as the phenomenon in which local failure is followed by Allen and Schriever 1972 collapse of adjoining members which in turn is followed by further collapse and so on, so that widespread collapse occurs as a result of local failure. A progressive collapse is characterized by the loss of load-carrying capacity of a relatively small portion of a structure due to an abnormal load which, in turn, triggers a cascade of failure Gross and McGuire 1983 affecting a major portion of the structure. Progressive collapse is a situation in which a localized failure in a structure, caused by abnor- mal load, triggers a cascade of failure affecting a major portion of the structure. Progressive collapse occurs when an initiating localized failure causes adjoining members to Smilowitz and Tennant 2001 be overloaded and fail, resulting in an extent of damage that is disproportionate to the originat- ing region of localized failure. When an initiator event causes a local failure in a building, the resulting failure front will prop- Hansen et al. 2005 agate throughout the structure until the failure front is arrested, or until the remaining structure becomes geometrically unstable. Progressive collapse occurs when local failure of a primary structural component leads to the Khandelwal and El-Tawil 2005 failure and collapse of adjoining members, possibly promoting additional collapse. A progressive collapse is a catastrophic partial or total collapse that initiates from local struc- Ellingwood 2005 tural damage and propagates, by a chain reaction mechanism, into a failure that is dispropor- tionate to the local damage caused by the initiating event. A catastrophic partial or total structural failure that ensues from an event that causes local Ellingwood and Dusenberry 2005 structural damage that cannot be absorbed by the inherent continuity and ductility of the struc- tural system A progressive collapse initiates as a result of local structural damage and develops, in a chain Ellingwood 2006 reaction mechanism, into a failure that is disproportionate to the initiating local damage. Progressive collapse of building structures is initiated by the loss of one or more load-carrying members. As a result, the structure will seek alternate load paths to transfer the load to struc- tural elements, which may or may not have been designed to resist the additional loads. Failure Mohamed 2006 of overloaded structural elements will cause further redistribution of loads, a process that may continue until stable equilibrium is reached. Equilibrium may be reached when a substantial part of the structure has already collapsed. The resulting overall damage may be disproportion- ate to the damage in the local region near the lost member. “Disproportionate collapse” is structural collapse disproportionate to the cause. In structures susceptible to this type of collapse, small events can have catastrophic consequences. Dispro- Nair 2006 portionate collapse is often, though not always, progressive, where “progressive collapse” is the collapse of all or a large part of a structure precipitated by damage or failure of a relatively small part of it. Progressive collapse is characterized by a distinct disproportion between the triggering spa- Starossek 2006 tially-limited failure and the resulting widespread collapse. Review on Quantitative Measures of Robustness for Building Structures Against Disproportionate Collapse 153

Appendix A. Continued. Progressive collapse, where the initial failure of one or more components results in a series of Canisius et al. 2007 subsequent failures of components not directly affected by the original action is a mode of fail- ure that can give rise to disproportionate failure. Disproportionate collapse results from small damage or a minor action leading to the collapse of a relatively large part of the structure. Progressive collapse is the spread of damage through Agarwal and England 2008 a chain reaction, for example through neighboring members or storey by storey. Often progres- sive collapse is disproportionate but the converse may not be true. Progressive collapse is a failure sequence that relates local damage to large scale collapse in a Krauthammer 2008 structure. A disproportionate (or progressive) collapse of a structure is one that initiates from local dam- age and, rather than being arrested by the capability of the structural system to redistribute Ellingwood 2009 forces and bridge around the damaged area, propagates to a final damage state that involves a major portion of the structure. one or several structural members suddenly fail, whatever the cause accident or attack. The Menchel et al. 2009 building then collapses progressively, every load redistribution causing the failure of other structural elements, until the complete failure of the building or of a major part of it. Disproportionate collapse: A collapse that is characterized by a pronounced disproportion between a relatively minor event and the ensuing collapse of a major part or the whole of a Starossek and Haberland 2010 structure. Progressive collapse: A collapse that commences with the failure of one or a few structural components and then progresses over successively affected other components. A disproportionate (or progressive) collapse of a structure is initiated by local damage, that Xu and Ellingwood 2011 cannot be contained and that propagates throughout the entire structure or a large portion of it, to the point where the extent of final damage is disproportionate to the initiating local damage. Progressive collapse of a building can be regarded as the situation where local failure of a pri- Kokot and Solomos 2012 mary structural component leads to the collapse of adjoining members and to an overall dam- age which is disproportionate to the initial cause. Progressive collapse is a chain reaction mechanism resulting in a pronounced disproportion in Parisi and Augenti 2012 size between a relatively minor triggering event and resulting collapse, that is, between the ini- tial amount of directly damaged elements and the final amount of failed elements. Progressive collapse of a structural system occurs when the local failure of a single primary Fallon et al. 2016 load-bearing element or a small group of elements triggers a larger, more widespread collapse of adjoining portions of the structure. Progressive collapse is described as building collapse caused by the loss or failure of a struc- tural load-bearing member because of load hazards. Localized failure facilitates load redistri- Nazri et al. 2017 bution to the adjacent member, which then initiates partial or total progressive collapse of a building. Progressive collapse of structures is the phenomenon of an initial local failure mushrooming to Pantidis and Gerasimidis 2017 the global level, resulting in the stiffness degradation of a relatively large part of the structure. Eventually, partial or total collapse of the structure occurs. Progressive collapse, a chain reaction or disproportionate propagation of failures following Xiao and Hedegaard 2018 damage to a relatively small portion of a structure. Progressive collapse is a collapse that begins with localised damage to a single or a few struc- Adam et al. 2018 tural components and develops throughout the structural system, affecting other components. the global disproportionate failure/collapse of a structure due to the initial local damage of a Feng et al. 2020 single or a few structural components triggered by some specific accidental events. If a structure lacks sufficient robustness, local damage induced by accident events, such as Kong et al. 2020 blasts, impacts, fires, or a combination of such things could spread widely to the remaining parts of the structure, leading to a complete or disproportionate collapse. 154 Jian Jiang et al. | International Journal of High-Rise Buildings

Appendix B. List of definitions of robustness Source Definition of robustness The ability of a structure to withstand events like fire, explosions, impact or the consequences Eurocode 1 Part 1-7, 2006 of human error without being damaged to an extent disproportionate to the original cause. Ability of a structure or structural components to resist damage without premature and/or brit- GSA tle failure due to events like explosions, impacts, fire or consequences of human error, due to its vigorous strength and toughness. The robustness of a system is defined as the ratio between the direct risks and the total risks (total risks is equal to the sum of direct and indirect risks), for a specified time frame and con- JCSS 2008 sidering all relevant exposure events and all relevant damage states for the constituents of the system. the ability of an engineered structure or system that enables it to survive a potentially damag- IstructE 2002 ing incident or extreme event without disproportionate loss of function. JSSC 2005 The state of being strong, tough and rigid. Robustness is broadly recognized to be a property which can not only be associated with the structure itself but must be considered as a product of several indicators: risk, redundancy, duc- tility, consequences of structural component and system failures, variability of loads and resis- Faber 2006 tances, dependency of failure modes, performance of structural joints, occurrence probabilities of extraordinary loads and environmental exposures, strategies for structural monitoring and maintenance, emergency preparedness and evacuation plans and general structural coherence. Insensitivity to local failure is referred to as robustness. Robustness is a property of the struc- Starossek 2006 ture alone and independent of the cause and probability of initial local failure. Robustness refers to the manner in which certain performance objectives or system properties are affected by hazardous or extreme conditions. It makes no sense to speak of a system being Maes et al. 2006 robust without first specifying both the feature and the perturbations of interest. Robustness is concerned with the “how and to what extent” these specified performance objectives are affected by the specified perturbations. Robustness is the ability of a structure to avoid disproportionate consequences in relation to the Agarwal and England 2008 initial damage. Structural robustness can be viewed as the ability of the system to suffer an amount of damage Biondini et al. 2008 not disproportionate with respect to the causes of the damage itself. The robustness of a structure, intended as its ability not to suffer disproportionate damages as a Bontempi et al. 2007 result of limited initial failure, is an intrinsic requirement, inherent to the structural system organization. The notion of robustness is that a structure should not be too sensitive to local damage, what- Vrouwenvelder 2008 ever the source of damage. Insensitivity of a structure to initial damage. A structure is robust if an initial damage does not Starossek and Haberland 2010 lead to disproportionate collapse. As the capacity of a structure to withstand damages without suffering disproportionate Brando et al. 2012 response to the triggering causes while maintaining an assigned level of performance. Ability of a structure in withstanding an abnormal event involving a localized failure with lim- Brett and Lu 2013 ited levels of consequences, or simply structural damages. Robustness is accomplished when the structure response is proportioned to the actions applied Formisano et al. 2015 to it. These actions could appear in different ways, e.g. loads exceeding the design ones, acci- dental loads or damage to members. The concept of structural robustness is generally associated with the ability of a structural sys- tem to resist widespread collapse or failure as the result of an initial perturbation. This pertur- Fallon et al. 2016 bation can be manifested as a change in the applied load (attributable to an extreme event), a change in the structure’s capacity (because of damage), or both. Building robustness to progressive collapse can be generally defined as a measure of the ability Li et al. 2019 of a system to remain functional in the event of the local failure of a single component or mul- tiple connected components.