Assessment of robustness of structures: Current state of research

Abstract

The concept of structural robustness and relevant design guidelines have been in existence in the progressive collapse literature since the 1970s following the partial collapse of the Ronan Point apartment building; however, in the more general context, research on the evaluation and enhancement of structural robustness is still relatively limited. This paper is aimed to provide a general overview of the current state of research concerning structural robustness. The focus is placed on the quantification and the associated evaluation methodologies, rather than specific measures to ensure prescriptive robustness requirements. Some associated concepts, such as redundancy and vulnerability, will be discussed and interpreted in the general context of robustness such that the corresponding methodologies can be compared quantitatively using a comparable scale. A framework methodology proposed by the authors is also introduced in line with the discussion of the literature.

Keywords

structural robustness / abnormal exposure / vulnerability / collapse / consequence

Introduction

The fundamental characteristics of a structure are conventionally described by stiffness, strength, ductility, as well as stability. These properties can generally be controlled through codified design procedures to meet specific requirements. However, during the long service life of a structure it could be exposed to some exceptional events which are outside the coverage of a normal design process. Examples of such events include failure of a steel connection because of growing fatigue fracture and a sudden loss of a load carrying component due to impact or explosive loading. These events are typically unpredictable and the cause is difficult to control, therefore it is not feasible nor practical, and clearly not economical, to include such effect directly into design considerations. A more rational approach would to ensure that the structure can withstand such an exposure without the so-called disproportionate damage or collapse. The ability of the structure to do this emphasizes a new dimension in the spectrum of the characteristics of a structure, namely the structural robustness.

Abnormal exposures, whether accidental or due to malicious acts, and the resulting damaged state often impose unforeseen demands and unanticipated reductions in capacity. Older structures were able to accommodate some of these unforeseen situations as a result of less efficient construction techniques and imprecision, and therefore conservatism, in the design process. Modern design eliminates many of these undocumented factors of safety while accommodating ever-greater challenges of architectural expression; as a result, modern building design may be more vulnerable to unforeseen conditions during the life of a building. Consequently it is understandable that the concept of structural robustness has attracted increasingly more interest in the structural engineering research and design communities.

Structural robustness is a general term that may be used to refer to a variety of desirable structural behaviors in a variety of working conditions that is not explicitly designed for through a standard code-required design procedure. In the context of this study, the definition is confined to the ability of a structure in withstanding an abnormal event involving a localized failure with limited levels of consequences, or simply structural damages.

An evaluation of the structural robustness thus requires at least the following concepts and corresponding models:

1) Definition of abnormal event set - in a simple case this may be removal of a load carrying element, targeted or random.

2) Definition of limited levels of consequences - the ultimate level is the complete collapse of the system, but in a comprehensive evaluation this is expected to cover a range of levels, similar but not necessarily equivalent to the performance limit states. A framework of limiting criteria is required.

3) dDefinition of a unified robustness measure - this should allow for benchmarking, cross-comparison, and the possibility of optimisation.

A large body of studies exist in the literature on relevant topics, mostly associated with progressive collapse and some deal with the assessment of robustness from a reliability perspective. However, there is no single framework that incorporates all the essential aspects in an explicit, transparent and quantitative manner leading to a comprehensive outcome in terms of quantification of the structural robustness and systematic design requirements to achieve a specified robustness target.

This paper is aimed at providing a comprehensive account with regard to the current state of research in this subject area. Recognizing the fact that progressive collapse has developed into a dedicated area of robustness, and as such has become a specialized subject, the topic of resistance to progressive collapse will be discussed only briefly in the last part of the paper for the sake of completeness. The focus is placed on the more general aspects of the structural robustness.

Basic concepts and terminology

Structural robustness is generally concerned about the ability of a system to withstand abnormal circumstances without disproportionate failure. The abnormal circumstances could arise from extreme events such as explosions, impact, fire, or the consequences of human errors and structural deterioration [1]. Structural robustness has been recognized as an intrinsic requirement which is fundamentally inherent to the structural system organization [2] and is associated with the vigorous strength and toughness [3]. A range of variants in the more detailed definitions exist. Slotine and Li [4] defines structural robustness as the degree to which a system is insensitive to effects outside the design considerations. Beeby [5] regards robustness as a specified energy absorption capacity of the structure.

Although there is a lack of generally accepted methods for the direct quantification of structural robustness, various proposals on the definitions of some closely associated characteristics exist. Lind [6] proposed the definition of vulnerability as the ratio of failure probability of the damaged system to that of the undamaged system, and such a definition of vulnerability may be easily converted to a measure of robustness. Augusti et al. [7] uses the concept of sensitivity (in damage terms) of the facility to a given event. Hendawi and Frangopol [8] looked into the reliability of redundant systems using a failure path approach that requires all of the failure probability to be enumerated. Ellingwood and Leyendecker [9] were among the first to advocate the alternative path analysis which involves removing a member to determine if the “damaged” structure can tolerate the redistribution of loads. Agarwal et al. [10] developed a so-called “rings & rounds” approach to evaluate the vulnerability of structural systems.

Ultimately, a systematic quantification of system robustness needs to be assessed in the context of three fundamental elements, 1) type of abnormal exposure (abnormal “hazard”), 2) the structural consequence of such exposure, and 3) the broader consequence or risk including fatalities and economic loss. While defining the exposure and assessing the broader consequence will involve a number of other factors (see a comprehensive list in Baker et al. [11]), the structural consequence under a given exposure lies at the center of the whole framework and generally would be the most controllable aspect as far as structural engineers are concerned. In this paper we shall confine ourselves only to the robustness in relation to the structural consequence under a given exposure in the form of local failure or severe damage causing a serious disruption to the structural system.

Closely related to the above-outlined concept of structural robustness is the broader-sense structural redundancy which forms the basis for the system to adapt to the structural change, and the ductility which determines whether the system can sustain the usually large deformations without progressive loss of strength in the course toward the establishment of a new equilibrium state. With this in mind, it is useful to clarify several concepts that may be involved in the different approaches of assessing the robustness of a structure.

Robustness - there is no absolute universally accepted definition of robustness. However, while the wording may vary, the underlying theme or concept is relatively consistent, and several common keywords can be extracted from the various definitions such as damage, vulnerability, disproportion, consequences, insensitivity, unforeseen loading, risk and so on. While robustness itself may be considered to be a property inherent to a particular structure, it is a function of other structural properties including strength, stiffness, ductility, as well as the structural organization and redundancy.

Consequences are the potential outcomes of events. These can be considered in terms of loss of life, economic costs and damage to the environment (JCSS 2008) [12]. As far as structural robustness is concerned, it is the indirect consequences, or the subsequent additional damage following a direct consequence of an exposure, that is of interest. For example, in the event a structure is exposed to collision leading to failure of a column (direct consequence), if the column failure led to the progressive collapse of the structure, then collapse would be categorised as an indirect consequence.

Damage- a common damage definition in the context of evaluating redundancy or robustness is the complete removal of one, or sometimes a few, members from a structure. This simplified definition represents the ultimate direct consequences in many abnormal exposure scenarios, and therefore allows the analysis to focus on the indirect consequence regardless the cause. An alternative to removing an entire member is to introduce damage to a member which is a proportion of the cross section area. Such damage can represent section loss due to corrosion for example [8].

Redundancyis a concept which is closely related to robustness. In the extreme, a system is completely non-redundant when the failure of a component equals the failure of the entire system [13]. To quote Ghosn and Moses [14], “redundancy is defined as the capability of the structure to continue to carry loads after the failure of one main member.” The implied capacity certainly requires adequate static redundancy but involves a number of other contributors, particularly in terms of deformability and ductility. Different types of redundancy may also be considered such as internal redundancy (member redundancy), structural redundancy and load path redundancy [15].

Static Determinacy- While redundancy is related to static indeterminacy it has been demonstrated that the two should not be considered to be equivalent [16]. Lind (1995) [6] notes that redundancy is related to static determinacy, although only superficially. Additionally, higher levels of static indeterminacy can even reduce the collapse loads of trusses and frames and decrease the reliability of a structure [17].

Vulnerability- JCSS [12] differentiates between vulnerability and robustness based on their relationship to consequences; vulnerability is regarded as being related to the direct consequences of damage whereas robustness is related to the indirect consequences of damage.

When vulnerability is confined to being a measure related to the tolerance of a structural system to an initiation exposure, it becomes generally a term reciprocal to robustness. Lind [6] discusses vulnerability in tandem with damage tolerance. They are considered to be complementary concepts as a system which is vulnerable is not damage tolerant (and hence not robust) and vice versa. Agarwal et al. [18] adopted a similar approach. A structure is considered to be vulnerable if damage from any exposure results in consequences which are disproportionate to the original damage event. It is proposed that system vulnerability is related to the form of the structure. Disproportionate consequences arise from a structure which has poor form and connectivity when renders it susceptible to progressive collapse.

Reliabilityin the context of structural robustness brings aspects of uncertainties into the framework. The reliability is classically evaluated by reliability indices which can be determined for individual members or the overall system; however a member-orientated approach could lead the design of non-robust systems [19]. Schafer and Bajpai [20] note that current design methods provide estimates of the failure probability (P_f) for each component; however no direct estimate of the system P_f is used. Furthermore, knowledge of the sensitivity of P_f of a damaged building is believed to be a key quantity regarding decision making for catastrophic unforeseen events.

With respect to robustness and progressive collapse the probability of failure is commonly considered in terms of conditional probabilities as follows [12,21]:

(1)P(F)=P(F|D{H}_i)P(D|H_i)P(H_i),

where F = event of structural collapse, P(H_i) = probability of hazard H_i, P(D|H_i) = probability of local damage, D, given that H_i occurs, and P(F|DH_i) = probability of collapse, given that the hazard and local damage both occur.

While P(F|DH_i) is concerned with system behavior, P(D|H_i) is related to individual member behavior. Therefore, in contrast with P(H_i), they are both within the remit of the structural engineering design [22]. Typically methods proposed to assess redundancy or robustness focus on P(F|DH_i) by assuming a specific level of damage such as the failure of a member. This facilitates an evaluation of the system response.

Risk- The risk, R, associated with a particular event is equal to the probability, P, of the event occurring, multiplied by the consequence, C, of the event [12]. Therefore, using the equation for P(F) in Eq. (1), the risk can be evaluated as follows;

(2)R=P(F|D{H}_i)P(D|H_i)P(H_i)\times C_i,

where C_i is the consequence of event i. Given that preventing damage in the event of a hazard may be unfeasible, the above equation is sometimes simplified as follows [23]:

(3)R=P(F|D{H}_i)P(H_i)\times C_i.

Damage Effort- As discussed previously, structural robustness is concerned with disproportion between an instance of damage and the consequences of damage. Solely considering the consequences is not sufficient to determine whether this disproportion exists. The probabilistic approach detailed above is one method to evaluate disproportion. Another method explored in the literature is to also incorporate damage effort into the analysis. There are a number of methods which have been proposed for this purpose. Lu et al. [24] and the other papers from this group suggest that the damage effort is proportional to the loss of principal stiffness due to a damage event. Smith [25] uses the energy required to cause failure of a member to quantify damage effort.

Assessment of robustness: General methodologies

A conceptual generic model

In spite of the various approaches to describing robustness, it is generally understood that the robustness of a structure should measure how (in) sensitive the structural system properties can be to an abnormal exposure, such as loss of a load carrying member in structure terms. Herein we shall propose a generic model with which various approaches with connections to the above defination may be looked upon from a unified perspective.

Let G denote a global system property, and X be a generic system variable against which the “abnormal exposure” may be measured. The exposure may therefore be defined as a disturbance to X, i.e., \delta X, and the system sensitivity to the exposure can subsequently be written as

(4)S={\displaystyle \frac{\delta G}{\delta X}}.

S varies in (0, ∞). The system robustness (insensitivity), can then be expressed as

(5)R={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+S}},

such that for extremely sensitive systems, S approaches infinity, R approaches zero; whereas, for perfectively insensitive systems, S approaches zero, R approaches 1. The general relationship between R and S is illustrated in Fig. 1.

When X becomes a discrete variable and cannot be differentiated (or simply a comparative measure does not exist), which would indeed be the case in many abnormal exposure scenarios for example the notional removal of a column, Eq. (1) can be modified such that the sensitivity is represented by a normalized system property change, as

(6)\overline{S}=\delta \overrightarrow{G}.

\overline{S}varies in (0,1), with zero being no change and 1 being 100% change. Thus, the system robustness becomes:

(7)R=\mathrm{1}-\overline{S}.

It can be seen in subsequent sections that a variety of approaches may be fitted into the above general formulation one way or another.

A risk-oriented conceptual model

Baker et al. [11] propose a conceptual model that relates robustness to both direct and indirect risks. This approach divides consequences into 1) Direct consequences associated with element damage (which could be considered as proportional damage), and 2) Indirect consequences assimilated with subsequent structural failure (which could be considered disproportionate damage).

An index is formed by comparing the risk associated with direct and indirect consequences.

The index of robustness I_ROB is defined as

(8)I_{\mathrm{R}\mathrm{O}\mathrm{B}}={\displaystyle \frac{R_{\mathrm{D}\mathrm{i}\mathrm{r}}}{R_{\mathrm{D}\mathrm{i}\mathrm{r}}+R_{\mathrm{I}\mathrm{n}\mathrm{d}}}},

where R_Dir is the direct risk and R_Ind is the indirect risk.

The risks are defined as illustrated in Fig. 2. First, an exposure occurs which has the potential of damaging structural elements in a system; this is named the exposure before damage, or EX_BD. If no damage occurs (\overline{D}), then the analysis is finished. If damage does occur, a number of damage (D) states can follow. For each of these damaged states, there is a probability that system failure (F) occurs. Consequences are assimilated with each of the possible damage and failure scenarios, and are specified as direct (C_Dir) or indirect (C_Ind).

Provided that the required direct and indirect risks (consequences) are available, I_ROB can be evaluated and it takes values between zero and one, with robust systems having an index value approaching one. In principle, such a definition could potentially account for the effect of repair strategies, inspection and maintenance as well as the system’s ability to accommodate accidental events, since such actions can alter failure consequences and this risk.

We can re-write Eq. (8) into the following generalized form:

(9)I_{\mathrm{R}\mathrm{O}\mathrm{B}}={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+R_{\mathrm{I}\mathrm{n}\mathrm{d}}/R_{\mathrm{D}\mathrm{i}\mathrm{r}}}}={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+{\overline{R}}_{\mathrm{I}\mathrm{n}\mathrm{d}}}},

where {\overline{R}}_{\mathrm{I}\mathrm{n}\mathrm{d}} may be regarded as the system “sensitivity” in risk terms, and the formula converges with the generic expres sion in Eq. (5). Such generalisation allows the robustness index to be evaluated even in the absence of a direct risk measure such as in the case of a definite loss of a structural member.

Static stiffness based approaches

Static stiffness based methods, by which the damage resilience of a structure may be evaluated, have been proposed by a number of research papers in one form or another. In general stiffness matrices can be used to assess the stability of a structure following a damage event.

Nafday [26] provides an informative and concise discussion regarding system safety performance metrics for skeletal structures which are based on properties of stiffness matrices. Two indices are proposed to evaluate the stability of a structure. The first index, which is based on the stiffness matrix condition number, is a measure of the distance of a structural stiffness matrix from a set of noninvertible matrices, which represent unstable structural states. The second index uses the determinant of the static stiffness matrix to assess the stability of a structural system. The determinant provides a measure of the linear dependence of column vectors in the stiffness matrix, with increasing linear dependence tending toward instability. An unstable system will have a zero determinant. The index is simply the normalized determinant of the elastic stiffness matrix;

(10)\mathrm{\Delta }s=\left|K_{\mathrm{n}}\right|,\mathrm{0}\le \mathrm{\Delta }s\le \mathrm{1.}

While both of these indices are useful from the point of view of structural system stability they are not directly evaluating robustness. However another index is also presented which compares the normalized determinant of the intact structure with the normalized determinant of a damaged state;

(11)I=\left|K_{\mathrm{n}}\right|/\left|K_{\mathrm{n}}*\right|,

where \left|K_{\mathrm{n}}\right| and \left|K_{\mathrm{n}}^{}\right| are the determinants of the normalized stiffness matrix of the intact and damage structures respectively. More critical members will have a higher importance factor, I. This index can be considered to have more in common with the objectives of robustness assessment as it provides an evaluation of how the structure withstands a damage event.

In a unified context as elaborated in Section 3.1, a normalized sensitivity in terms of the determinant of the stiffness matrix may be written as:

(12)\overline{S}=\delta {\overline{K}}_{\mathrm{n}}=\left|K_{\mathrm{n}}\right|-\left|K_{\mathrm{n}}^{}\right|.

Then Eq. (7) is equally applicable here for the evaluation of the robustness index in terms of the determinants of the stiffness matrix. The soundness of the so-derived robustness then depends on how well the determinants of the stiffness represent the change of the system property in question, and it needs to be calibrated.

Similarly, Starossek and Haberland [27] considers the static stiffness matrix in order to define a stiffness based robustness index;

(13)R_{\mathrm{s}}=\min \left({\displaystyle \frac{\det K_{\mathit{i}}}{\det K_{\mathrm{o}}}}\right),

where det(K_i) and det(K_o) are the determinants of the stiffness matrices for the damaged and intact structures respectively.

Several other papers from Bristol University (e.g., Lu et al. [24]) propose a methodology to determine a vulnerability index, which relates to the form of the structure. Disproportionate consequences arise from a structure which has poor form and connectivity which renders it susceptible to progressive collapse. Vulnerability, which essentially is a system sensitivity measure according to the definition given in Section 3.1, is quantified as follows [28]:

(14)\varphi ={\displaystyle \frac{\gamma }{D_r}},

where the vulnerability ϕ is the ratio of the separateness (damage consequences), γ, to the damage effort, D_r.

The separateness quantifies the deterioration in the structural form following a damage event, is a defined as follows;

(15)\gamma ={\displaystyle \frac{Q(S)-Q(S\prime )}{Q(S)}},

where Q(S) and Q(S′) are the well-formedness of the intact and damaged structures respectively. The well-formedness is a stiffness based measure of the quality of the form and connectivity of the structure. The damage effort, D_r, is proposed to relate to stiffness via the loss in principal stiffness caused by a damage event.

Equation (15) is effectively a normalized measure of system sensitivity in terms of the well-formedness. Recall Eq. (7),

(16)R=\mathrm{1}-\overline{S}=\mathrm{1}-\gamma ={\displaystyle \frac{Q(S\prime )}{Q(S)}}.

Methods such as those outlined above, are purely evaluating the form and stability of a structural system. Therefore they are independent of both loading and boundary conditions. One potential advantage of this approach is that the susceptibility to any arbitrary damage event may be considered without the normal constraints which would govern a response based analysis. Furthermore, they are easy to implement as they do not require any computationally intensive structural response analyses. However, as noted by Starossek and Haberland [29], there may be little correlation between a stiffness robustness analysis and a traditional strength based analysis. Therefore, while there may be some complementary benefit, methods which only consider the form and stability of a structure cannot feasibly replace traditional response based analysis.

Energy based approaches

Several authors have employed the principles of energy absorption and energy balance in order to evaluate progressive collapse and robustness. General discussions of energy based approaches can be found in Arup [30] and England and Agarwal [31].

Smith [25] proposed a methodology by which energy principles can be used to assess the collapse resistance of a structure. The primary aim of this method is to evaluate critical sequences of damage events which lead to collapse of a structure. The most critical sequences are those with the lowest damage effort. Damage effort, also referred to as work to failure, and the consequences of a damage event are expressed in terms of strain energy.

The failure sequences are evaluated solely in terms of the total work to failure. Consequently the method is limited to collapse sequences. In such cases the consequences of each collapse sequence are the same and as a result the sequences may reasonably be evaluated in terms of their respective energy requirement. However, for individual members or sequences which do not lead to failure, evaluation in terms of damage effort alone is not useful. It may however be useful to evaluate individual members or non-collapse sequences both in terms of effort and consequences using an energy ratio such as that proposed by Smith [25] or the vulnerability index described by Pinto et al. [28].

The method seeks to find the damage sequences with the lowest energy requirement. In some cases it is likely that these sequences will be the same as the natural failure paths of the structure. When this is not the case, however, the minimum energy sequences represent somewhat arbitrary collapse sequences with respect to the real expected structural behavior. Prior to the onset of progressive collapse, each event in the sequence is essentially an independent failure event requiring an external energy input. Given that sequences may be arbitrary, they cannot for example be attributed to overloading of the structure or indeed a collapse sequence which is precipitated by the failure of a single member. Such arbitrary sequences raise the question of how the required input energy for each failure event arises, in particular for sequences in which several failure events are required before structural collapse, and also the validity of identifying arbitrary failure sequences as the most critical.

In the present study we have explored the methodology on representative structures. Herein a 2-D truss, shown in Fig. 3, is briefly illustrated with a linear elastic analysis. As the members only experience axial forces, this greatly simplifies the calculation of the work to failure for the members. It is assumed that buckling does not occur. Therefore the work to failure is the same for members in tension and compression.

Table 1 shows the work to failure of each member. As members only transmit axial force, the work to failure is a function of the member area and member length.

The method allows the failure sequences with the lowest work to failure to be determined. A summary of the five most critical collapse sequences is detailed below. The most critical sequence is sequence “5-3” as it has the lowest energy requirement. “5-3” indicates that member 5 is the first member in the sequence, followed by member 3.

In this example, we also identify the failure sequences in terms of their status. “Natural” means that the failure sequence accords with the sequence which will naturally occur in the structure given that the first member has failed. “Arbitrary” describes the situation in which, given that the first member has failed, the second member does not accord with the natural failure path. For this example, in which failure of two members is sufficient to cause failure of the structure, there is a good correlation between the most critical failure sequences and the natural failure paths in the structure. However, for structures with collapse sequences of several members, such agreement is less probable.

The following energy ratio was proposed to identify the most critical member at each stage of the analysis;

(17){\rho }_{\mathrm{e}}={\displaystyle \frac{\mathrm{\Delta }{W}_{\mathrm{f}\mathit{i}}}{\mathrm{\Sigma }{U}^i}},

where \mathrm{\Delta }{W}_{\mathrm{f}\mathit{i}} is the net work of failure for member i and \Sigma {\mathit{U}}^{\mathit{i}} is the consequence of removing member i. \Sigma {\mathit{U}}^{\mathit{i}} is defined as the total strain energy in the structure after the failure of member i. It is noted that this energy ratio can be linked to the generic sensitivity-oriented robustness definition given in Section 3.1, via a transformation of the vulnerability index of Eq. (17):

(18)S={\displaystyle \frac{{\displaystyle \sum U^i}}{\mathrm{\Delta }{W}_{\mathrm{f}i}}}={\displaystyle \frac{\mathrm{1}}{{\rho }_{\mathrm{e}}}}.

A proposed framework for a comprehensive quantification of robustness

Individual property such as strength or stiffness based indices typically evaluate the respective property of a post damage residual structure with respect to the intact structure (e.g., Ghosn and Moses [14]; Frangopol and Curley [16]). In contrast energy absorption methods are primarily concerned with the ductility of a structure. Considering the objective of quantifying structural deterioration, Brett and Lu [32] propose a more comprehensive index which, by incorporating additional structural properties, provides an enhanced evaluation of the consequences arising from a damage event, namely the elastic stiffness, the yield strength and the ductility.

The diagram in Fig. 4 shows schematically the load displacement relationships for an intact state and three alternative damaged states of a notional structure. These are damage states which might arise after a damage event such as failure of a member. Damage consequences (or system change as referred to in the present paper), which are evaluated with respect to the intact structure, may take a variety of different forms depending upon the underlying properties of the structure. Damage state 1, for example, only exhibits a reduction in ultimate strength relative to the intact state. Damage state 2 experiences a reduction in elastic stiffness and yield strength. Damage state 3 exhibits a reduction in elastic stiffness, yield strength, ultimate strength and also ductility.

For any given residual state, deterioration may have occurred in one or more of these characteristic properties. Each individual property provides a quantification of a different aspect of the structural deterioration. To fully describe structural changes, each of the above four structural properties may be combined. To this end, the authors propose a general framework by which each property can be incorporated into a comprehensive unified evaluation of the post-damage structural deterioration.

A consequence index (effectively a normalized system property change) is defined for each structural property. The indices range from zero to one, such that a value of zero indicates that no deterioration has taken place and a value of one indicates collapse of the structure. The stiffness consequence index is as follows:

(19)C_{\mathrm{k},i}=\mathrm{1}-{\displaystyle \frac{k_{\mathrm{i}}}{k_{\mathrm{o}}}},

where C_{\mathrm{k},i} is the stiffness consequence index, k_i is the elastic stiffness of damage state i, k_{\mathrm{o}} is the elastic stiffness of the intact. Similarly, the consequence indices for yield strength, ultimate strength and ductilty (C_{u,f,i},, C_{\mathrm{u}\mathrm{f},i} and C_{\mathrm{u},\mathrm{d},i} respectively) are obtained by relating the residual and intact structural properties. The indices are then combined to create a single index.

The combination of the indices is carried out using the following approach in each case; C_{\mathrm{s},i}, C_{\mathrm{u},i} and C_i . For example, the index C_{\mathrm{s},i} is determined as follows;

(20)C_{\mathrm{s},i}=C_{\mathrm{k},i}+C_{\mathrm{y},i}-C_{\mathrm{k},i}\times C_{\mathrm{y},i}

where C_{\mathrm{k},i} is the stiffness consequence index and C_{\mathrm{y},i} is the yield strength consequence index. This approach fulfils the objective of preserving a range of zero to one, where a value of one indicates collapse.

Using the concept of sensitivity, the above structural consequence or system change index is effectively the normalized sensitivity, \overline{S}, which can be converted to the robustness index by Eq. (7).

The results displayed in Table 3, determined using a finite element analysis of a 2D truss, illustrate the potential pitfalls of only using strength to evaluate structural deterioration relative to the intact structure. An analysis of two damage states is presented. Each is determined by removing a member from the structure and using the consequence indices to quantify the system change.

In each damage state the reduction in ultimate strength is equivalent. This is indicated by the strength consequence index in the shaded (C_u,f,i) column. However, the other consequence indices clearly demonstrate that damage state B exhibits a significantly greater reduction in stiffness, yield strength and ductility. Evidently, with respect to these damage states, the ultimate strength alone does not provide a representative quantification of the general structural deterioration.

Combining each structural property into a single index is useful from the point of view of simplicity, clarity and also completeness. This addresses many of the general robustness assessment requirement discussed by Starossek and Haberland [29]. Moreover, the combined index can be easily degenerated if, given the requirements of a specific analysis, it is desirable to focus on a single property such as strength or ductility.

Assessment of robustness: Reserve factor/redundancy approaches

Wisniewski et al. [33] presented a simple procedure for robustness quantification based on load-capacity evaluation for existing railway bridges. The procedure is essentially derived from the methodology proposed earlier by the senior author of the above paper in Ghosn and Moses [14]. Three key states are defined, namely 1) functionality, 2) ultimate, and 3) damaged condition limit state. The definition of a damage condition limit state provides a more flexible term, allowing for specialization to, for example, removal of a load carrying member in a frame system, and removal of a few reinforcing bars in the case of a deck-girder bridge.

The procedure then evaluates the load reserve factors and subsequently the redundancy at both local (member) and system levels, using a finite element model. Live load is specified, in the railway bridge case considered as a passing train. Load factor is defined by the multiplier of a trainload until a specified target state is reached.

At the member level the member strength and the design-required member capacity (considering partial safety factor), R and R_req, respectively, are first calculated (appropriate nonlinear analysis methods or FE, and conforming relevant codes such as Eurocode). R_req is determined for the most critical member in a structure. Load factors corresponding to reaching the R and R_req are calculated, denoted as LF₁ and LF_req, respectively. A member reserve factor is simply calculated as

(21)r_{\mathrm{1}}={\displaystyle \frac{L{F}_{\mathrm{1}}}{L{F}_{\mathrm{r}\mathrm{e}\mathrm{q}}}}.

This reserve factor effectively represents the strength reserve of the intact structure with respect to the design strength requirement.

At the system level, the load factor to reach the functionality limit state, for example globally a 1/500 mid-span deflection and locally a limit maximum strain, LF_f, is obtained through the bridge FE analysis. The functionality reserve ratio is simply calculated as

(22)R_{\mathrm{f}}={\displaystyle \frac{L{F}_{\mathrm{f}}}{L{F}_{\mathrm{1}}}}.

It should be noted that the above reserve factor is with respect to the intact strength load factor, and therefore is only dependent upon the system properties regardless the design load level.

Similarly the system reserve ratio for the ultimate limit state and damaged condition limit state can be calculated. The ultimate limit state is defined by a global (deflection) criterion and a local (strain) criterion. The damage condition is defined by a removal of a critical member or reduction of member strength (e.g., through removal of part of main rebars), and the limit state may be defined by a separate set of global and local criteria, for example not to cause collapse.

(23)R_{\mathrm{u}}={\displaystyle \frac{L{F}_{\mathrm{u}}}{L{F}_{\mathrm{1}}}},

(24)R_{\mathrm{d}}={\displaystyle \frac{L{F}_{\mathrm{d}}}{L{F}_{\mathrm{1}}}}.

The above reserve ratios are normalized with respect to the corresponding target ratios (based on empirical and reliability considerations- this is where inherent uncertainties may be accounted for, similar to the concept of specifying partial safety factors, thus save a full reliability-probabilistic analysis in a routine robustness procedure), to become a “redundancy” measure (in a sense of redundant capacity), as

(25)r_{\mathrm{f}}={\displaystyle \frac{R_{\mathrm{f}}}{R_{\mathrm{f},\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}}},

(26)r_{\mathrm{u}}={\displaystyle \frac{R_{\mathrm{u}}}{R_{\mathrm{u},\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}}},

(27)r_{\mathrm{d}}={\displaystyle \frac{R_{\mathrm{d}}}{R_{\mathrm{d},\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}}}.

As proposed in Ghosn and Moses [14] R_f,target = 1.1, R_f,target = 1.3, R_f,target = 0.5.

Finally through the incorporation of the basic strength reserve factor for the intact structure, the redundancy index is evaluated with respect to the specific design requirement as

(28){\phi }_{\mathrm{r}\mathrm{e}\mathrm{d}}=\min (r_{\mathrm{1}}{r}_{\mathrm{f};}{r}_{\mathrm{1}}{r}_{\mathrm{u};}{r}_{\mathrm{1}}{r}_{\mathrm{d}}).

Note that

(29)r_{\mathrm{1}}{r}_{\mathrm{f}}={\displaystyle \frac{L{F}_{\mathrm{1}}}{L{F}_{\mathrm{r}\mathrm{e}\mathrm{q}}}}\times {\displaystyle \frac{L{F}_{\mathrm{f}}}{L{F}_{\mathrm{1}}}}\times {\displaystyle \frac{\mathrm{1}}{R_{\mathrm{f},\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}}}={\displaystyle \frac{L{F}_{\mathrm{f}}}{L{F}_{\mathrm{r}\mathrm{e}\mathrm{q}}}}\times {\displaystyle \frac{\mathrm{1}}{R_{\mathrm{f},\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}}}={\displaystyle \frac{R_{\mathrm{f},\mathrm{r}\mathrm{e}\mathrm{q}}}{R_{\mathrm{f},\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}}}.

Similarly,

(30)r_{\mathrm{1}}{r}_{\mathrm{u}}={\displaystyle \frac{R_{\mathrm{u},\mathrm{r}\mathrm{e}\mathrm{q}}}{R_{\mathrm{u},\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}}},

(31)r_{\mathrm{1}}{r}_{\mathrm{d}}={\displaystyle \frac{R_{\mathrm{d},\mathrm{r}\mathrm{e}\mathrm{q}}}{R_{\mathrm{d},\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}}},

It is worth noting that for the damage condition limit state evaluation step, a series of damage conditions may be considered, for example removal of each and every individual member, one at a time. Each such damage condition will give rise to one independent system reserve ratio, and the minimum is considered as the overall R_d. The whole set of R_d’s may be employed in ranking the importance of individual members. This then converges with the approach discussed earlier in Section 3.5.

Frangopol and coworkers [16,34] investigate the influence of damage on redundancy and system reliability. It is demonstrated, using two simple truss examples, that static indeterminacy is not a suitable measure of system strength and that structural redundancy should be considered with respect to individual member behavior and overall system strength. The paper discusses different deterministic definitions of redundancy including:

The reserve redundancy factor:

(32)R_{\mathrm{1}}={\displaystyle \frac{L_{\mathrm{int}act}}{L_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}}},

the residual redundancy factor:

(33)R_{\mathrm{2}}={\displaystyle \frac{L_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{d}}}{L_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}}}},

(33)the strength redundancy factor:

(34)R_{\mathrm{3}}={\displaystyle \frac{L_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}}}{(L_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}}-L_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{d}})}}.

where L_intact denotes the ultimate load factor (or simply ultimate strength) of the undamaged structural system, L_design denotes the design load factor (or a nominal applied load), and L_damaged is the ultimate load factor (or ultimate strength) of the damaged structural system.

R₃ measures the relative change of the damaged structure with respect to the intact structure in terms of strength and is dependent only upon the structure itself. R₃ is actually another form of measuring the magnitude of the strength change, and we can re-write it as

(35)R_{\mathrm{3}}={\displaystyle \frac{\mathrm{1}}{(\mathrm{1}-L_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{d}}/L_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}})}}={\displaystyle \frac{\mathrm{1}}{(\mathrm{1}-R_{\mathrm{2}})}}.

This definition of redundancy may be interpreted as the reciprocal of the sensitivity of the structure to a specified damage, with the sensitivity being measured as the normalized change in strength of the damaged structure with respect to the intact structure. Thus when the sensitivity approaches zero, i.e., the system strength is not changed with the given damage, the structure is said to have infinite redundancy with respect to the given damage, and vice versa.

The capacity of the strength redundancy factor (R₃) to provide a measure of structural safety and the influence of the damage of different member is demonstrated using a set of examples subjected to different types of damage. The results indicate the importance of the combination of damaged members in a multiple damage scenario and also the influence on the failure type (ductile or brittle) on the system strength.

The same framework of evaluating the R factors has also been also extended to incorporate probabilistic redundancy measures [16,34]. In this case the load and strength of the members are assumed to be random variables. A reliability index is determined for each member in a structure according to the different damage levels.

Reliability index:

{\beta }_{\mathrm{3}}={\displaystyle \frac{{\overline{M}}_i}{\sigma (M_i)}},

where

(36)M_i=S_i-Q_i.

M_i = performance function of element i, S_i = random strength of element i and Q_i = random load effect on member i.

Additionally the influence of the failure type (brittle or ductile) is demonstrated by plotting the reliability index against the mean applied load for each damage level. The probabilistic redundancy in systems is expressed as

(37){\beta }_R={\displaystyle \frac{{\beta }_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}}}{{\beta }_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}}-{\beta }_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{d}}}}.

We can also re-write Eq. (36) to yield:

(38){\beta }_R={\displaystyle \frac{\mathrm{1}}{\mathrm{1}-({\beta }_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{d}}/{\beta }_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t}})}},

which is analogous to the deterministic Eq. (35) albeit using reliability indices.

Additional measures of the redundancy in terms of the reliability indices of the intact and damaged structures may also be constructed [35].

The generality of the above formulation of redundancy quantification (which herein has been interpreted to connect to the robustness) permits optimisation to be sought in terms of the system redundancy. Frangopol and Klisinski [36] investigated the effect of optimisation on the redundancy on a three-dimensional space truss. Optimisation is considered in terms of minimising the weight of the truss subject to a number of constraints. Three redundancy and strength measures are considered; the reserve strength factor, the residual strength factor, and the redundancy factor. These measures are considered to be good indicators of the strength and redundancy as they are directly related to system strength for both the damaged and undamaged structure. When optimisation is introduced, in the form of weight minimisation, the material volumes of the truss are significantly reduced. This results in an increase in the number of fully stressed members at failure. The paper demonstrates a conflict between optimisation and the requirement for sufficient redundancy.

Assessment of robustness: Vulnerability and reliability approaches

Vulnerability is quantified as expressed in Eq. (14). Vulnerability is therefore a quantitative measure of the disproportion of a damage event. ϕ generally varies from zero to infinity.

Lind [6] considers the vulnerability as the reciprocal of damage tolerance and is defined as follows;

(39)v=v(r_{\mathrm{d}},S)={\displaystyle \frac{P(r_{\mathrm{d}},S)}{P(r_{\mathrm{o}},S)}},

where v is vulnerability, r_d and r_o represent the damaged and intact states respectively and S is the prospective loading. So, P(r_d,S) is the probably of failure of the damaged condition given the prospective loading S. v varies in a range of 1.0 to infinity.

Obviously either of the above definitions of vulnerability may be converted to a robustness measure, which should generally be inversely related to the vulnerability (in a similar fashion as to the sensitivity). We can come up with the following transformation relationships:

R_{\mathrm{o}}={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+\varphi }};\varphi =(\mathrm{0},\infty )and{R}_{\mathrm{o}}=(\mathrm{0},\mathrm{1})

or

R_{\mathrm{o}}={\displaystyle \frac{\mathrm{1}}{V}};V=(\mathrm{1},\infty )and{R}_o=(\mathrm{0},\mathrm{1}).V=(\mathrm{1},\infty )

The trend of the above relationships is essentially the same as that illustrated in Fig. 1 (Section 3.1).

It is interesting to note that JCSS [12] defines vulnerability the other way round, as

(40)I_{\mathrm{V}}={\displaystyle \frac{R_{\mathrm{D}}}{V_{\mathrm{A}}}},

where the vulnerability I_v is equal to the ratio of the direct risk R_D, to the parameter which is used to measure the value of the indirect risk. The value might be expressed in terms of lost lives, monetary value or another suitable measure. The implication from this equation is that when a minimum direct risk results in a maximum indirect risk, the vulnerability is the smallest, which tends to contradict the general understanding in the context of structural robustness under abnormal conditions.

Structural robustness under progressive collapse scenarios

The partial collapse of Ronan Point in 1968 was widely regarded as a starting point for the structural engineering community to become aware of the need for robustness against disproportionate failure, which is often of a progressive collapse scenario. This incident prompted the introduction of regulations for designing against disproportionate collapse in the 5th Amendment to the Building Regulations in 1970 in the UK, and numerous research studies followed. A new wave of attention to this subject was brought about following the 1995 bombing of the Alfred P. Murrah Federal Building, the bombing of the US embassy buildings in two African countries in 1998, and the collapse of the World Trade Centre Towers in New York on September 11, 2001.

A progressive collapse is a catastrophic partial or total structural failure that ensues from an event that causes local structural damage that cannot be absorbed by the inherent continuity and ductility of the structural system [21]. Progressive collapse is the spread of damage through a chain reaction, for example through neighboring members or storey by storey. There are a number of different types of progressive collapse; pancake-type collapse, zipper-type collapse, domino-type collapse, section-type collapse, instability-type collapse and mixed-type collapse [37]. Such characterization of collapse is useful from the point of view of identifying links between collapse types and structural configuration, which may provide a more informed platform from which to consider appropriate collapse prevention mechanisms.

Progressive collapse is thus closely associated with both redundancy and robustness. However, the concept of structural robustness in the context of resisting progressive collapse has been expressed in the form of specific limiting requirements, rather than a quantifiable property of the structure that may be measured and compared. In this respect, research studies on the structural robustness against progressive collapse have been focused on two areas of the subject, namely the “indirect design” and the “direct” design approaches.

The so-called “indirect design” approach resorts to specifying minimum levels of robustness through prescriptive measures of tie forces, continuity and ductility to develop resistance to disproportionate failure. The provision of ties increases the structural continuity, resulting in inbuilt redundancy to redistribute loads in case a part of the structure is removed accidentally. The tie force methodology is event independent and is intended to provide a minimum level of accidental damage tolerance [38]. In practice the ties are usually provided by one or a combination of the following: 1) the (steel) members including the connections that must be capable of transferring the horizontal tying forces. 2) The steel bar reinforcement- provided that it is anchored to the steel frame and embedded in concrete. 3) The steel mesh reinforcement in a composite slab with profiled steel sheeting.

The “direct design,” on the other hand, involves specific analysis and design procedures to ensure a structure’s ability to absorb local damage and resist disproportionate failure. There are generally two possible means to achieve this: 1) Alternative path method, which emphasizes the behavior of the structure after damage has occurred, regardless of the action causing damage, and relies on the ductility and continuity of the structure to redistribute forces. In this approach, it is typically assumed that part of the structure is lost, and design is carried out to enable the remaining structure to redistribute the loads to the undamaged areas. BS5950 recommends that the structure should be designed to bridge over a loss of un-tied member and that the area of collapse be limited and localized. This is usually achieved by removing each un-tied element, one at a time, and checking that on its removal the area of structural risk of collapse is within a specified limit. 2) Local resistance or key element method: this approach is recommended especially for situations where the loss of an element cannot be tolerated or when tying or bridging over of a member is not possible.

Stemming from the Fifth Amendment [39], more detailed requirements and design provisions have been gradually incorporated in various design codes and guides, e.g., the Building Regulations 1985 (Approved Document A) [40], BS EN 1991-1-7 [1], GSA [3], UFC 4-023-03 [41], 2009, and ASCE 7-05 [42]. A comprehensive review of the evolution of the design provisions as well as the development of relevant analysis methods against disproportionate collapse can be found in a report by Arup [30].

Concluding remarks

Robustness of structures in withstanding abnormal exposures involving local failure is an important consideration in the design and management of modern civil engineering structures. It effectively constitutes a new dimension of the fundamental structural capacities, to add onto the conventional stiffness, strength and ductility.

Research on the subject of structural robustness, particularly in terms of generalized characterization and quantification, is still limited; however, conceptual methods and specific measures have been proposed from a diverse range of evaluation perspectives.

In a unified context, several existing methods may be associated with the quantification of structural robustness. In this paper, these approaches are classified into three groups, namely 1) general methodologies, including consequence-based and generalized capacity (involving strength, ductility and energy) methodologies, 2) load capacity or redundancy based approaches, and 3) vulnerability and reliability based approaches. As it stands, each different method has been proposed to measure the structural capacity in withstanding certain abnormal exposures from different perspectives, and the metric used is not directly comparable. However, as demonstrated in the paper, it is possible to transform different measures into a unified scale, and this allows a direct comparison and benchmarking of different measures for structural robustness.

In the context of general robustness as discussed in this paper, structural robustness concerning resistance to progressive or disproportionate collapse, especially those induced by removal of a vertical load carrying member, is a specialized theme. With this in mind, readers are directed to the progressive collapse literature regarding the evolution of the robustness-oriented design guides and corresponding structural analysis techniques under progressive collapse scenarios.

References

[1] BS EN 1991-1-7:2006: Eurocode 1: Actions on structures- Part 1-7: General actions: Accidental actions. London: British Standards Institute (BSI), 2006

[2] Bontempi F, Giuliani L, Gkoumas K. Handling the exceptions: Robustness assessment of a complex structural system. In Structural Engineering, Mechanics and Computation 3, 2007, 1747-1752

[3] General Services Administration (GSA). Progressive Collapse Analysis and Design Guidelines for New Federal Office Buildings and Major Modernisation Projects. Washington D C, 2003

[4] Slotine J, Li W. Applied Nonlinear Control. Englewood Cliffs (NJ): Prentice Hall, 1991

[5] Beeby A W. Safety of structures, and a new approach to robustness. Structural Engineer, 1999, 77(4): 16-21

[6] Lind N. A measure of vulnerability and damage tolerance. Reliability Engineering & System Safety, 1995, 48(1): 1-6

[7] Augusti G, Borri C, Niemann H J. Is Aeolian risk as significant as other environmental risks? Reliability Engineering & System Safety, 2001, 74(3): 227-237

[8] Hendawi S, Frangopol D M. System reliability and redundancy in structural design and evaluation. Structural Safety, 1994, 16(1-2): 47-71

[9] Ellingwood B, Leyendecker E. Approaches for design against progressive collapse. Journal of the Structural Division, 1978, 104(3): 413-423

[10] Agarwal J, Blockley D, Woodman N. Vulnerability of structural systems. Structural Safety, 2003, 25(3): 263-286

[11] Baker J, Schubert M, Faber M. On the assessment of robustness. Structural Safety, 2008, 30(3): 253-267

[12] Risk Assessment in Engineering. Principles, System Representation and Risk Criteria. Zurich: Joint Committee on Structural Safety (JCSS), 2008

[13] Bertero R, Bertero V. Redundancy in Earthquake-Resistant Design. Journal of Structural Engineering, 1999, 125(1): 81-88

[14] Ghosn M, Moses F. NCHRP Report 406: Redundancy in Highway Bridge Superstructures. Washington D C: Transportation Research Board, 1998

[15] Connor R, Dexter R, Mahmoud H. NCHRP Synthesis 354: Inspection and Management of Bridges with Fracture-Critical Details. Washington D C: Transportation Research Board, 2005

[16] Frangopol D, Curley J. Effects of damage and redundancy on structural reliability. Journal of Structural Engineering, 1987, 113(7): 1533-1549

[17] Sebastian W. Collapse considerations and electrical analogies for statically indeterminate structures. Journal of Structural Engineering, 2004, 130(10): 1445-1453

[18] Agarwal J, Blockley D, Woodman N. Vulnerability of 3-dimensional trusses. Structural Safety, 2001, 23(3): 203-220

[19] Ghosn M, Moses F, Frangopol D M. Redundancy and robustness of highway bridge superstructures and substructures. Structure and Infrastructure Engineering, 2010, 6(1): 257-278

[20] Schafer B, Bajpai P. Stability degradation and redundancy in damaged structures. Engineering Structures, 2005, 27(11): 1642-1651

[21] Ellingwood B R, Dusenberry D O. Building design for abnormal loads and progressive collapse. Computer-Aided Civil and Infrastructure Engineering, 2005, 20(3): 194-205

[22] Starossek U, Haberland M. Measures of Structural Robustness-Requirements & Applications. In Proceedings of the Structures Congress. Reston, ASCE, 2008, 1-10

[23] Decò A, Frangopol D. Risk assessment of highway bridges under multiple hazards. Journal of Risk Research, 2011, 14(9): 1057-1089

[24] Lu Z, Yu Y, Woodman N J, Blockley D I. A theory of structural vulnerability. Structural Engineer, 1999, 77(18): 17-24

[25] Smith J W. Energy approach to assessing corrosion damaged structures. Proceedings of the Institution of Civil Engineers, Structures and Buildings, 2003, 156(2): 121-130

[26] Nafday A M. System Safety Performance Metrics for Skeletal Structures. Journal of Structural Engineering, 2008, 134(3): 499-504

[27] Starossek U, Haberland M. Evaluating Measures of Structural Robustness. In: Proceedings of the Structures Congress. Reston, ASCE, 2009, 1-8

[28] Pinto J T, Blockley D I, Woodman N J. The risk of vulnerable failure. Structural Safety, 2002, 24(2-4): 107-122

[29] Starossek U, Haberland M. Approaches to measures of structural robustness. Structure and Infrastructure Engineering, 2011, 7(7-8): 625-631

[30] Arup. Review of International Research on Structural Robustness and Disproportionate Collapse. London: Department for Communities and Local Government, 2011

[31] England J, Agarwal J. Recent developments in robustness and relation with risk. In: Proceedings of the ICE-Structures and Buildings. 2008, 161(4): 183-188

[32] Brett C, Lu Y. Developing a quantitative approach for assessment of structural robustness against collapse. In: Forde M C, ed. Structural Faults & Repair. Edinburgh: Engineering Technics Press, 2012

[33] Wisniewski D, Casas J R, Ghosn M. Load capacity evaluation of existing railway bridges based on robustness quantification. Structural Engineering International, 2006, 16(2): 161-166

[34] Frangopol D, Nakib R. Redundancy in highway bridges. English Journal, 1991, 28(1): 45-50

[35] Frangopol D, Iizuka M, Yoshida K. Redundancy measures for design and evaluation of structural systems. Journal of Offshore Mechanics and Arctic Engineering, 1992, 114(4): 285-290

[36] Frangopol D, Klisinski M. Weight-strength-redundancy interaction in optimum design of three-dimensional. Computers & Structures, 1989, 31(5): 775-787

[37] Starossek U. Typology of progressive collapse. Engineering Structures, 2007, 29(9): 2302-2307

[38] BS 5950-1:2000: Structural use of steelwork in buildings- Part 1: Code of practice for design- Rolled and welded sections. London: British Standards Institute (BSI), 2001

[39] The Building (Fifth Amendment) Regulations, 1970(S.I. 1970/109)

[40] The Building Regulations. Approved Document A: Structure. A3- Disproportionate Collapse. 1985 edition. Her Majesty’s Stationery Office (HMSO), 1985

[41] UFC 4-023-03: Design of Buildings to Resist Progressive Collapse. Washington D C: Department of Defence (DOD), 2005

[42] ASCE Standard No. 7-05: Minimum design loads for buildings and other structures. American Society of Civil Engineers (ASCE), 2005

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