Vertical collapse safety margin assessment for steel frames against earthquake-induced loss of column

Engineering Failure Analysis 105 (2019) 945–960

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Vertical collapse safety margin assessment for steel frames against earthquake-induced loss of column Jeriniaina Sitraka Tantelya, Zheng Hea,b, a b

T

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Dept. of Civil Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China

A R T IC LE I N F O

ABS TRA CT

Keywords: Collapse safety margin Vertical earthquakes Demand-to-capacity ratio Column loss Steel frame structures

A quantifiable approach measuring the seismic structural collapse under sudden column removal is proposed using the interaction of the vertical collapse margin ratio (CMRV), the demand-tocapacity ratio (DCR) and the robustness (R). The assessment of CMRV needs the identification of the vertical maximum considered earthquakes. A classification of the ground motions based on the vertical-to-horizontal (V/H) ratios was used to obtain the vertical design spectrum. Six schemes of steel structures with various scenarios of sudden column removal were investigated to extract CMRV, DCR, and R. Then, a nonlinear regression analysis allows finding a general formulation of CMRV. A 15-story steel structure validates the proposed formulation of the CMRV. The two paths for estimating the minimum CMRV (CMRV, min) were the use of the recommended value of DCR and the minimum value of R inside the approximated CMRV expression, and the probabilistic distribution of CMRV, min from the analyses of different beam deformation states. The study of eighteen archetypes with 119 scenarios of sudden column removal reveals the value of the CMRV decreases when the V/H or/and DCR increase. The new maximum values of DCR with respect to R and its precise ranges are found with the use of the contour curves of CMRV, min.

1. Introduction Over the past decades, efforts at raising awareness, promoting prevention, preparedness, and mitigation have been stepped up around the world in reducing the cost from natural disasters, blasts or structural collapse. Governmental documents, such as UFC [1] and GSA [2], provide design requirements necessary to reduce the potential for progressive collapse in the structures that experience severe localized damage, e.g. column loss. In the field of earthquake engineering, FEMA [3] aims to minimize the risk of structural collapse under the maximum considered earthquake (MCE) and defines the safety in terms of CMR. Since then, studies about the CMR have gained more attention. It involves the modification of CMR [4,5], the minimum acceptable value of CMR [6], the risk assessment [7], the estimate of earthquake loss [8] and the corroboration of factors inside a design code [9]. CMR becomes a valid and reliable structural safety indicator. Although these documents focus on the topic of structural collapse, no other connection seems to exist between them. On one hand, the CMR indicates only the safety in the lateral direction of the structures, and on the other hand, the recommended procedures in UFC [1] and GSA [2] do not relate to the phenomenon of structural collapse but rather focus on preventing its occurrence. Therefore, an opportunity to relate these two different fields of investigation about the structural collapse appears. This combination can improve the understanding of the design against progressive collapse, and can extend the application of the conventional CMR in the vertical direction of the structure.

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Corresponding author at: Dept. of Civil Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China. E-mail address: [email protected] (Z. He).

https://doi.org/10.1016/j.engfailanal.2019.07.049 Received 18 June 2018; Received in revised form 7 July 2019; Accepted 18 July 2019 Available online 23 July 2019 1350-6307/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Failure Analysis 105 (2019) 945–960

J.S. Tantely and Z. He

Hazards, such as deliberate attacks, vehicle impact, natural causes, etc., cause the disruption in vertical integrity of the structure. Due to the limited database of the collapse events from these hazards, it is not possible to assess the probability of occurrence for a specific hazard or group of hazards. Thus, the methodology against progressive collapse is threat-independent, and focuses on the structural effect of the hazard only, i.e. the removal of load-bearing elements (wall, column, etc.). The guidelines incorporate redundancy into the progressive collapse resisting system to mitigate single points of failure and provide increased robustness (R) for extreme loading scenarios. Yet, they do not explicitly address R in the design. The redundancy is in the terms of DCR, which the guidelines recommend specific values regarding its upper limit [10]. DCR calculation comes from the alternate path method that requires the structure be able to bridge across the removed element. Although DCR below the recommended values assure the structural resistance against progressive collapse, there are two issues concerning the DCR, its lack of physical interpretation to predict structural collapse and its conservative values independent to the robustness of the structures. It is physically impossible to interpret the value of DCR when it is superior to the recommended ones, i.e. DCR value that implies structural collapse is undefined. For instance, the field experiment of Song and Sezen [11] reveals that even though four first-story columns were physically removed from the four-story building, and its DCR went above the recommended values, the damaged structure still stood. Weigand et al. [12] confirmed this inaccuracy of UFC and GSA recommended capacities or acceptance criteria through experimental data. The contributions from the infill walls and non-structural elements can be another reason for the standing of the damaged structure. If the structure remained in the elastic state, these secondary elements will add extra stiffness to the building, but under inelastic state (as it is the case in column loss) these infill walls would have already been destroyed due to their brittle material composition. Therefore, for the sake of simplicity, the calculation of the DCR neglects these secondary elements. It is also probable that the natural variability such as the usage of the building, or the change in loads can be a factor to keep a damaged structure standing. The DCR recommended values are fixed no matter the robustness of the structure. R can be defined as the insensitivity of a structure to initial damage, i.e. a structure is robust if initial damage does not lead to disproportionate collapse [13]. Baker et al. [14] used a risk-based index to estimate R. Calculability is challenging for such approaches. Fascetti et al. [15] established a damage-based measure for R, by determining the most critical sequence for successive removal of multiple columns and defining R as the minimum fraction of columns that must be removed to cause failure. Bao et al. [16] estimated R by removing a single column but considering multiple scenarios. The methodology determines the minimum R-value (R = 1) of the structures that indicates no collapse would occur under any of the sudden column loss scenarios, but it does not discuss its maximum value in order to have precise design objective for mitigating structural collapse. In addition, the methodology also permits to connect the DCR and R calculations, by evaluating DCR and R at the service-level gravity loading, and at the ultimate load intensity, respectively. Although DCR and R are related to the issue of progressive collapse, no meaningful interaction has been found between them yet. The use of CMR in the vertical direction of the structure is very promising in the context of establishing a probabilistic approach of quantifying progressive collapse. The only issue revolves around its applicability because an intact structure is very stiff along its height. However, any disruption of the vertical integrity of the structure weakens the building and makes it sensitive to seismic force. Estimating CMR in the vertical direction signifies the identification of the vertical maximum considered earthquake (MCEV). Several studies investigate the vertical ground motion using its horizontal counterpart. The horizontal-to-vertical ratio has a significant use in engineering seismology, especially characterizing site amplification [17], and classifying sites to use in ground motion models [18]. Researchers such as Ambraseys and Simpson [19], and Elnashai and Papazoglou [20] predicted the intensity of the vertical ground motion based on the variation of V/H to the site-to-source distance using several records from different earthquakes. In general, they concluded the vertical ground motion is stronger near the source of the earthquake, and bigger local magnitude (ML) gives bigger V/ H ratio. Bozorgnia and Campbell [21] contradicted the interaction ML-V/H but concurred with V/H ratio been sensitive to site-tosource distance. In conjunction with probabilistic seismic hazard assessment, researchers commonly developed the vertical design spectrum using a V/H ratio from ground motion prediction equations [22], or from seismological parameters of scenario earthquakes [23] to scale the horizontal spectrum. Kim et al. [24] formed the vertical design spectrum from statistical and regression analyses. Liu et al. [25] proposed a simple and reliable way to develop the vertical design spectrum by using the same formula, the same figure as the horizontal one based on the V/H ratio. The above methods paired with the loss of vertical integrity make the estimation of CMR feasible in the vertical direction. However, this CMR extension, called vertical CMR (CMRV), needs to be related to some established indicators about vertical collapses, such as DCR and R, in order to have a physical meaning. The aim of this paper is to explore the interaction of CMRV-DCR-R by assessing the CMRV of the steel structures under several scenarios of sudden column removal, by formulating this interaction using regression analysis. This proposed relation provides a new quantifiable way to determine structural collapse under sudden column removal by identifying the minimum CMRV (CMRV, min), and defines the maximum range of the values of DCR and R with the use of CMRV, min. 2. CMRV assessment 2.1. Important parameters for assessing CMRV The purpose of the CMR evaluation is to minimize the risk of structural collapse under the seismic load of the maximum considered earthquake (MCE). This structural collapse is often assumed to be at a predefined maximum lateral deformation. However, most cases of collapse in steel structures occur due to the reduction of their load-carrying capability with the weakening of the structural elements. This loss of load-carrying capability indicates that the gravity load plays an important role in the real collapse of the structure. On the other hand, the design to resist progressive collapse focuses on this gravity load. The methodology reduces the 946

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Fig. 1. The position of CMRV in preventing losses.

potential for the progressive collapse under sudden column removal by designing the structures to reach a certain magnitude and distribution of the demand on the structural elements with the DCR. Although both methodologies have a significant role in preventing losses before they happen, Fig. 1 presents a possible improvement in extending the use of CMR and reaching new insights in design against progressive collapse. CMRV is a probabilistic assessment of collapse risk due to ground motions in the vertical direction of the structures. Seismically designed steel structures rarely collapse vertically to strong earthquakes. For instance, the aftermath of the Northridge (1994) and Kobe (1995) earthquakes reveals that even though, strong vertical accelerations near the epicenter cause elements failure, most observed collapses are the results of large lateral displacement. The vertical collapse usually occurs when the structure losses its vertical resistance. It is difficult to point out precisely which removed member is most likely to trigger vertical collapse under earthquake. However, the two most recent guidelines [1,2] produced in the United States for progressive collapse mitigation use the sudden column loss as the principal design scenario. Although such a scenario happens under a gravity load, the procedure does capture the influence of a seism load - a dynamic phenomenon - occurring over the response time of the structure. The use of the horizontal and vertical components of the earthquake is physically correct, however, it blurs the boundaries between the conventional CMR and CMRV. This confusion can happen when the horizontal components are stronger than the vertical one and induce a collapse mechanism in the lateral direction, e.g. side-sway collapse that overshadows the vertical deformation due to sudden column removal. To reduce the complexity of the CMRV investigation, it is reasonable to assume that the conventional CMR comes from the use of the horizontal component of the earthquakes, while CMRV assessment uses the vertical one. The work of Khanmohammadi and Kharrazi [26] is an example of the feasibility of this assumption. However, in the present work, it is imperative not to detach completely both components because of their interaction, more specifically the V/H ratio, characterizes the earthquake itself. Therefore, the V/H ratio is maintained to differentiate each vertical ground motions and is used in the development of the vertical MCE and the proposed expression of the CMRV. The conventional CMR is the ratio of the intensity measure (IM) of 50% of the ground motions to the IM of the MCE. In one sense, CMR could be thought as a quantitative relation between the structural influence of a set of ground motions and a selected seism for the design. Analogically, CMRV could present a similar interpretation in its formulation. The methodologies involving the column removal have the necessary frameworks to investigate both general and local behaviors of the structure. In this regard, the structural robustness can represent the general standpoint of the CMRV since it measures the global strength of the structure under multiple column loss scenarios [16]. First, the methodology applies and holds constant the gravity loading G to the bays not directly affected by the column loss. Then, it applies the scaled gravity loading λG to the affected bays and increases the dimensionless factor λ from zero until the ultimate capacity (λ = λu) is reached. The robustness R of the structure is the minimum value of the normalized ultimate capacity λ over all columns removal scenarios. The local standpoint of the CMRV is the “key element” provisions applied to members whose column removal causes damage exceeding prescribed limits. DCR measures the damage of each structural element and has some recommended values for the design to reduce the potential for progressive collapse. DCR evaluation uses the unscaled gravity load, i.e. λ = 1. This investigation uses the DCR value of the beams to estimate the CMRV for the following reasons. 1) They are the primary elements that pick up the extra-loads. 2) Most investigation about a single column removal, the beams always turn out to be the “key element” of the structure. However, in the case of two or multiple column removals, the recommended DCR values of all the primary elements should be considered [11,27]. Therefore, the seismic (V/H and MCE) and gravity (λu, R, and DCR) related parameters are important in the CMRV investigation and formulation. However, the role of the V/H in the CMRV assessment requires further explanations. 2.2. Proposed classification of the vertical earthquakes The V/H ratio has practical importance in earthquake engineering, such as its use in developing the vertical design spectrum. One of the methods to obtain the vertical MCE is the use of the V/H ratio to reshape the horizontal one by using the same formula and the same figure but with different coefficients [25]. In this regard, the above horizontal-vertical relation is acceptable to a distance of 100 km without serious compromise, but should not be used beyond this distance without carefully considering the possible engineering consequences [28]. Therefore, for assessing the CMRV, 1636 earthquakes with 32,766 records are collected from the National Research Institute for Earth Science and Disaster Resilience [29] database verifying the following criteria: having ML > 2 947

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Fig. 2. V/H ratio for SSD ≤ 100 km.

with a source-to-site distance (SSD) equal to or < 100 km, and occurring between the year 1996–2016. Each ground motion has three records pointing to East-West (EW), North-South (NS), and Up-Down (UD). By using the average of peak ground acceleration (PGA) depending on the SSD, two curves of V/H ratios are obtained for different ML, as seen in Fig. 2. The overall relationship of the V/H ratio and the SSD is consistent with the model of Elnashai and Papazoglou [20]. However, the irregularity of the curves based on ML indicates that 1) ML has no significant influence on the V/H ratio whatever the SSD, which confirms the observation of Bozorgnia and Campbell [21]. 2) The scattering of the records along the V/H axis, which is particularly predominant for SSD < 30 km, shows a possible classification of the vertical ground motions based on the V/H ratio. Researchers statistically calculated the V/H ratio of the ground motions based on certain criteria. For instance, Newmark [30] identified V/H = 0.67 for the United States. Kawashima et al. [31] found V/H = 0.20 for Japan, which does not reflect the result in Fig. 2. Ambraseys and Simpson [32] gave a value of 1.75 for a worldwide V/H ratio but only considered a SSD < 15 km and ML > 6. Therefore, it cannot fully represent the collected records in this work. Mohammadioun and Mohammadioun [33] found a V/H = 0.75 from near source earthquakes. Finally, Elnashai [34] determined a V/H = 1.00 from 130 records. Based on the suggested V/H ratios in the literature and the upper and lower bounds of the average V/H in this work, three specific values (0.50, 0.75, and 1.00) are identified. Most of the records with SSD > 20 km spread between V/H equal to 0.50–0.75. Records with SSD < 20 km have their average V/H above 0.75–1.00, and the remaining records are part of V/H < 0.50. Note that, some of the collected records have a V/H > 1.00 if treated individually, which is the actual procedure in the CMR assessment. Thus, by focusing on the maximum V/H of the earthquakes, four categories (C) emerge, as seen in Fig. 3 with C0 (29.505%), C1 (32.13%), C2 (22.24%), and C3 (16.125%). On the other hand, if the classification considered both V/H ratios of the earthquakes, the coupling effect between twodimensional components needs to be addressed, which is an issue outside the concern of this paper. To avoid the issue of which value of V/H ratio should represent each V/H category, the CMRV assessment uses the V/H ratios that separate the categories (0.50, 0.75, and 1.00). This allows estimating directly the limit values of CMRV for each V/H category. In another word, both lower and upper bounds of C1 and C2 will be defined, while C0 and C3 will have their upper and lower bounds, respectively, because both extremities of C0 and C3 are difficult to estimate due to the unpredictable nature of the earthquake. Therefore, thirty vertical ground motions are selected from three categories (C1, C2, and C3) that are above the V/H of interest. The selection prioritizes the records with maximum V/H regarding their category (see Table 1). Note that NIED [29] identifies their records with a name combining the station code, the year, the date, and the exact time of the earthquake. For example, the record AIC0010102230723 means AIC001/01/0223/0723 (station/2001/February 23rd/07:23). 2.3. Performance analyses of the archetypes To assess the CMRV and the important parameters for its later approximation, a set of archetype that gives a wide range of values each variable is needed. A 4-story by 3-bay steel frame structure from Tantely and He [35], designed using Sds (g) = 1.2653 and Sd1 (g) = 0.674 for the lateral load [36], acts as a model to create six different schemes for the CMRV evaluation. As the DCR of concern is from the beam elements, a sequence of two lengths of span (5–6 m) is utilized to differentiate one scheme from another, as shown in Fig. 4. In addition, three beam cross-sections – HEB300, HEB330, and HEB360–are respectively used to obtain three different

Fig. 3. A proposed classification of vertical earthquakes using the maximum V/H ratio. 948

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Table 1 Selected ground motions for the categories C1, C2, and C3. Category

Nb

Name

Year

ML

Maximum V/H

C1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

AIC0010102230723 AIC0011112142319 AIC0020110252012 AIC0030102230723 AIC0030109221811 AIC0040603160624 AIC0049712140534 AIC0050104032357 AIC0090110270305 AIC0110307090214 AIC0120106011116 AIC0120106031133 AIC0130106010041 AIC0130106031133 AIC0140106011116 AIC0170111170534 AIC0170907270944 AIC0179703161451 AKT0010808161720 AKT0170806140852 AKT0190806140930 AKT0190806150231 AKT0190806182355 AOM0030909100039 AOM0031110071151 AOM0040109172039 AOM0060808161720 AOM0090203022039 AOM0100808161720 AOM0101312131325 AIC0120109221811 AIC0120612180017 AIC0180102230723 AKT0180112022202 AOM0110109172039 CHB0020106250127 CHB0020111170132 CHB0049608160103 CHB0060111170132 CHB0140111170132 CHB0150111170132 FKS0071307040006 FKS0091103191856 FKS0110106111322 FKS0121504161638 GIF0070805121147 GNM0011507292135 HKD0760605120109 HKD0779609280807 HKD0910011152319 HKD0910912280913 HKD0949905130259 HYG0080101140855 IBR0011110260208 IBR0061504161638 IBR0119708090534 IBR0129612211523 IBR0131109271256 IBR0149606030218 IBR0151110100657

2001 2011 2001 2001 2001 2006 1997 2001 2001 2003 2001 2001 2001 2001 2001 2001 2009 1997 2008 2008 2008 2008 2008 2009 2011 2001 2008 2002 2008 2013 2001 2006 2001 2001 2001 2001 2001 1996 2001 2001 2001 2013 2011 2001 2015 2008 2015 2006 1996 2000 2009 1999 2001 2011 2015 1997 1996 2011 1996 2011

4.9 3 3.7 4.9 4.2 4 3.9 5.1 3.4 4.3 4 4.2 4.8 4.2 4 3.8 4 5.8 4 4 4 4 4 4 5 4.1 4 4.8 4 4 4.2 3 4.9 6.4 4.1 4 4.4 4.4 4.4 4.4 4.4 5 6.1 4.4 4 4 4 5 4.4 4.5 5 6.4 4.1 5 4 4.7 3.8 4 3.7 4

0.645 0.749 0.712 0.526 0.624 0.745 0.75 0.571 0.7 0.744 0.553 0.682 0.666 0.555 0.626 0.721 0.652 0.742 0.566 0.745 0.745 0.581 0.707 0.658 0.703 0.66 0.566 0.697 0.723 0.544 0.99 0.997 0.815 0.865 0.967 0.918 0.94 0.795 0.93 0.881 0.987 0.957 0.858 0.995 0.96 0.966 0.799 0.995 0.999 0.992 0.991 0.927 0.998 0.991 0.872 0.998 0.998 0.846 0.936 0.961

C2

(continued on next page)

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Table 1 (continued) Category

Nb

Name

Year

ML

Maximum V/H

C3

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

AIC0010603160624 AIC0019609082316 AIC0020603160624 AIC0029609082316 AIC0030603160624 AIC0040106210834 AIC0040109221811 AIC0040907270944 AIC0050106210834 AIC0090111170534 AIC0110412312330 AIC0120412312330 AIC0129909241433 AIC0150110111338 AIC0150208110756 AIC0170106031133 AIC0179712140534 AKT0169608110312 AKT0170112022202 AKT0189710270407 AKT0199710270407 AKT0230806150225 AOM0061207042229 AOM0081207042229 AOM0110502122123 AOM0110906150332 AOM0119908030032 AOM0119909130532 AOM0140107180747 AOM0140808161720

2006 1996 2006 1996 2006 2001 2001 2009 2001 2001 2004 2004 1999 2001 2002 2001 1997 1996 2001 1997 1997 2008 2012 2012 2005 2009 1999 1999 2001 2008

4 3.5 4 3.5 4 3.8 4.2 4 3.8 3.8 2.9 2.9 3.8 3.8 4.3 4.2 3.9 5.9 6.4 5.1 5.1 4 4 4 4 4 4.5 4 4.3 4

2.365 1.576 1.66 1.621 2.3 1.103 1.565 2.155 1.172 1.155 1.754 1.683 1.31 2.745 2.24 1.793 2.3 3.049 1.599 2.493 1.576 1.466 1.282 1.457 2.671 1.588 3.546 1.33 1.499 2.052

Fig. 4. Structural schemes and column removal considered for evaluating CMRV.

structural capacities against the column removal. The columns have HE cross-sections and the letter “r” with a subscript number designates the considered column for the number of removal scenarios in a scheme. The “line-element” approach was used to model all the schemes as seen in Fig. 5 [37]. This approach, used in OpenSees, is rapid and efficient for analytical simulations [38]. For the column removal scenarios, the column element formed by the nonlinearBeamColumn element and two slave nodes at its extremities are removed to keep the beam-column connection intact as suggested in GSA and UFC procedures. Further design properties of the steel structures are available in Tantely and He [35]. Following the GSA guidelines for nonlinear dynamic procedures, G is applied on the structure and then followed by the sudden column removal and the application of the listed vertical ground motions in Table 1. As expected, the dominant collapse mechanism 950

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Fig. 5. OpenSees modeling method.

is the beam yielding-type. However, some removal cases, such as r2, r3, r5, and r9, reveal that the beam yielding collapse mode can be replaced by the inelastic buckling failure of the column at the lower floors, as observed by Gerasimidis and Sideri [39]. This change of collapse mechanism increases the DCR value of one of the columns on the ground floor and makes it the “key element” of the structure. Therefore, as shown in Table 2, the outcomes of the analyses (CMRV, DCR, λu, and R) are from the removal cases that have the maximum DCR value belonging to a beam element. Moreover, the collapse criterion used for the CMRV estimation is the recommended rotation (θ = 21%rad) for the steel beams in GSA [10]. 3. CMRV formulation 3.1. Limitation of data The data from the CMRV assessment come from six schemes of steel structures subjected to 119 scenarios of column removal in structures. The use of these simple models made the analyses controllable and time-efficient. Although the aim to cover a big range of value for each considered variables is achieved, it is possible different models might present different outcomes. The prospect of dissimilar results could arise, for example, from the presence of seismic control devices or an asymmetric or irregular scheme that influence the behavior of the structure under sudden column removal. Due to the conservative definition of the collapse criterion, the range of the variables is likely to be minimum estimates of the distribution of structural collapse. The real collapse is still a complex subject to tackle. Nevertheless, the potential collapse of a structural element can be predicted with the use of empirically determined damage criteria. Precisely, the recommended rotation, used in the analyses, is the maximum allowable rotation limit for the beam element to restrain the possibility of collapse. This implies that the likelihood of the beam to collapse increases when its rotation goes above the recommended value. 3.2. Nonlinear regression equation for the CMRV The CMRV formulation should represent the forces that push the structure to collapse. The expression of the conventional CMR, i.e. the ratio of IMs, involves the ground motion alone, while CMRV has a vertical earthquake and gravity load. Therefore, based on the previous discussion regarding the important parameters of the CMRV formulation, a simplified equation resulting from the combination of two ratios gives a satisfactory way to approximate the CMRV. The first half represents the ratio from the influence of the seism force and the other half corresponds to the influence of the gravity load.

CMRV = β

IM λu IMMCEV DCR

(1)

where β is a structural variable combining the V/H and λu, and has the following expression.

β = (c1 V / H 2 + c2 V / H + c3 ) λ u (c4 V / H

c5 + c6)

+ (c7 V / H 2 + c8 V /H + c9)

(2)

And IM is the unit of the intensity measure. Table 3 gives the values of the coefficients c1–c9 for peak ground acceleration as intensity measure only. The coefficients c1-c3 and c7-c9 attenuate the downward and upward tendency of predictive CMRV curve, respectively. The CMRV-DCR relationship, Fig. 6, is the basis to evaluate the coefficients of the proposed CMRV because of the following reasons. 1) Even though the structure with a column loss is under two different loads, the influence of the gravity load is on the failed beam is more noticeable than the seismic one. 2) Among the parameters related to gravity loads, DCR is explicitly evaluated and checked for acceptance criteria in the design methodology against the progressive collapse. 3) The acceptance criteria for the 951

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Table 2 Summary of the results for all the archetypes with beams as key elements in DCR estimation. Scheme

Beam

S

HEB

1

300

330

360

2

300

330

360

3

300

330

360

4

300

330

360

Case

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Column removal

r1 r4 r6 r7 r8 r1 r4 r6 r7 r8 r1 r4 r6 r7 r8 r1 r4 r6 r7 r8 r1 r4 r6 r7 r8 r1 r4 r6 r7 r8 r1 r4 r6 r7 r8 r1 r4 r6 r7 r8 r1 r4 r6 r7 r8 r1 r4 r6 r7 r8 r1 r4 r6 r7 r8 r1 r4 r6 r7 r8

DCR

5.36 5.23 5.2 5.54 5.1 5.41 4.71 4.84 6.4 4.85 4.31 3.83 3.84 5.53 3.88 4.21 3.72 3.77 5.05 3.85 3.45 3.02 3.01 4.32 3.05 2.7 2.38 2.37 3.33 2.39 5.17 4.7 4.68 5.35 4.87 4.99 4.01 3.99 6.23 4.09 3.96 3.17 3.14 5.07 3.23 4.41 4.38 4.37 5.15 4.46 3.68 3.63 3.63 4.67 3.7 2.91 2.88 2.88 3.47 2.91

CMRV V/H = 0.50

V/H = 0.75

V/H = 1.00

1.73 1.88 1.92 1.78 2.17 1.7 2.57 2.42 1.4 2.39 2.77 2.92 2.92 1.79 2.87 2.78 2.95 2.93 2.31 2.88 3.02 3.12 3.12 2.76 3.12 3.16 3.27 3.2 3.04 3.27 2.12 2.58 2.61 1.92 2.36 2.16 2.87 2.85 1.52 2.86 2.87 3.09 3.1 2.25 3.05 2.74 2.35 2.75 2.13 2.72 2.96 2.97 2.88 2.63 2.96 3.12 3.13 3.1 3.02 3.12

1.39 1.51 1.52 1.32 1.53 1.37 1.93 1.78 1.18 1.78 2.01 2.06 2.03 1.32 2.04 2.02 2.08 2.06 1.61 2.05 2.14 2.18 2.18 2.01 2.18 2.22 2.26 2.23 2.15 2.26 1.5 1.94 1.96 1.39 1.78 1.68 2.03 2 1.22 2.02 2.03 2.17 2.17 1.58 2.16 2 1.78 2 1.51 1.99 2.08 2.09 2.04 1.96 2.08 2.15 2.19 2.15 2.13 2.19

1.27 1.36 1.38 1.19 1.46 1.25 1.7 1.62 1.11 1.62 1.73 1.78 1.75 1.19 1.77 1.73 1.78 1.78 1.57 1.78 1.8 1.87 1.87 1.73 1.86 1.88 1.89 1.85 1.81 1.89 1.41 1.7 1.71 1.3 1.62 1.59 1.76 1.72 1.13 1.75 1.76 1.84 1.85 1.55 1.83 1.73 1.62 1.73 1.42 1.72 1.79 1.79 1.72 1.71 1.78 1.85 1.88 1.84 1.8 1.88

λu

R

1.24 1.64 1.07 0.67 1.42 1.78 1.87 1.42 1.29 2.04 2.71 3.78 3.02 1.91 4.13 2.04 2.89 2.53 0.84 2.31 2.98 4.04 4.04 1.91 4 4.36 6.36 5.87 3.91 5.02 1.33 1.51 1.11 0.84 1.51 2 2.67 2.22 1.16 2.98 2.89 4.27 5.38 2.62 4.13 1.82 2.09 1.82 1.24 1.91 2.53 3.56 1.91 1.78 3.02 3.47 4.93 5.33 4.09 5.51

0.67

1.29

1.91

0.84

1.91

3.91

0.84

1.16

2.62

1.24

1.78

3.47

(continued on next page)

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Table 2 (continued) Scheme

Beam

S

HEB

5

300

330

360

6

300

330

360

Case

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119

Column removal

r1 r4 r6 r7 r10 r11 r13 r14 r15 r16 r1 r4 r6 r7 r10 r11 r13 r14 r15 r16 r1 r4 r6 r7 r10 r11 r13 r14 r15 r16 r1 r4 r7 r9 r11 r13 r14 r15 r16 r1 r4 r6 r7 r10 r11 r13 r14 r15 r16 r1 r4 r6 r7 r10 r11 r13 r14 r15 r16

DCR

5.13 4.12 4.53 3.69 4.53 3.72 5.32 4.71 3.79 5.01 4.49 3.36 3.57 2.97 3.55 2.96 5.75 3.62 3.02 4.22 3.42 2.61 2.7 2.35 2.67 2.32 4.38 2.73 2.36 3.25 5.37 4.35 4.28 5.44 4.28 5.53 5.08 4.38 5.12 5.03 3.56 4.33 3.53 4.45 3.54 6.23 4.42 3.61 4.52 3.81 2.8 3.35 2.81 3.35 2.81 4.87 3.36 2.83 3.33

CMRV V/H = 0.50

V/H = 0.75

V/H = 1.00

2.14 2.85 2.71 2.96 2.7 2.95 1.92 2.57 2.93 2.35 2.71 3.04 2.98 3.12 2.99 3.08 1.72 2.97 3.12 2.78 3.03 3.19 3.16 3.27 3.17 3.28 2.75 3.16 3.26 3.04 1.73 2.76 2.52 1.68 2.77 1.79 2.25 2.75 2.14 2.34 2.85 2.76 3 2.73 2.99 1.52 2.74 2.98 2.71 2.92 3.15 3.04 3.14 3.02 3.15 2.36 3.04 3.14 3.04

1.52 2.02 1.98 2.07 1.99 2.08 1.4 1.93 2.06 1.63 1.98 2.15 2.11 2.18 2.08 2.11 1.3 2.09 2.18 2.02 2.15 2.24 2.22 2.2 2.23 2.26 2 2.21 2.26 2.16 1.39 2 1.88 1.35 2.01 1.32 1.57 2 1.52 1.63 2.06 2 2.12 1.99 2.11 1.22 2 2.09 1.98 2.06 2.19 2.15 2.19 2.1 2.19 1.78 2.15 2.19 2.15

1.43 1.75 1.72 1.78 1.73 1.77 1.31 1.7 1.78 1.59 1.72 1.8 1.79 1.87 1.74 1.84 1.18 1.79 1.87 1.73 1.8 1.89 1.88 1.89 1.89 1.9 1.73 1.88 1.89 1.82 1.27 1.73 1.69 1.24 1.73 1.19 1.55 1.73 1.44 1.59 1.77 1.73 1.8 1.72 1.79 1.13 1.72 1.79 1.72 1.78 1.88 1.8 1.88 1.78 1.88 1.62 1.8 1.88 1.81

λu

R

0.98 2.09 2.09 2.8 2.04 2.8 0.49 0.89 2.53 0.8 2.27 2.93 3.56 3.91 3.2 2.84 1.29 3.56 4.04 2 3.6 4.53 4.84 5.38 6.62 3.87 3.07 5.73 6.49 3.24 1.24 1.87 2.44 1.07 2.44 0.71 1.51 1.16 0.98 1.96 2.58 2.93 3.6 1.82 3.96 1.42 2.76 2.71 1.78 3.16 3.56 4.49 5.11 4.31 4.62 2.22 4.53 5.96 4.22

0.49

1.29

3.07

0.71

1.42

2.22

recommended value of DCR consider the safety and the cost design of the structure. A structure with DCR far below the recommended value is very safe but not cost effective, while one with DCR far above is cheap and unsafe. Therefore, it is reasonable to constrain the proposed CMRV formulation to the unsafe part of this DCR spectrum since the structures above the recommended value are the most likely to collapse. Thus, in Fig. 6, the CMRV curves are drawn from the acceptance criterion of the DCR and above. The Pearson correlation 953

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Table 3 Value of the coefficients in β. Constant

Value

c1 c2 c3 c4 c5 c6 c7 c8 c9

5.616 −7.364 5.077 0.016 −3.771 −0.757 −3.636 5.912 −2.904

Fig. 6. Interaction of CMRV-DCR.

coefficient, ρ, assesses the relationship between two data by calculating the ratio of their covariance to their respective standard deviations. The values of ρ for the considered V/H ratios (ρ0.50, ρ0.75, and ρ1.00) are all over 0.90, which signify a strong uphill nonlinear relationship between the predicted and the observed data.

ρ (D, F ) =

COV(D, F ) σD σF

(3)

where COV is the coefficient of variation; D and F are the points from the observed and fitted data, respectively; σD and σF are the standard deviations of the observed and fitted data, respectively. The mean absolute error, MAE, for the overall CMRV prediction remains small. MAE is the average vertical distance between each point of the observed and predicted data.

MAE =

n

∑i =1

|Di − Fi|

(4)

where n is the number of the data point with DCR ≥ 3. 3.3. Validation of the proposed formulation To verify the proposed formulation of the CMRV, a high-rise 15-floor by 4-bay steel frame building is used (Fig. 7). This structure is a good model for the validation purpose in the event of sudden column removal because several elements will pick up and will redistribute the exceeding load. Moreover, the building size is more manageable for a nonlinear procedure using incremental dynamic analyses. The design of the frames follows the rules for earthquake resistance in the Eurocodes. Further detailed design information is provided in Gerasimidis et al. [40]. As the building has a symmetrical scheme, the considered cases for the sudden column removal are on the A and B axis and concern the floor number: 1, 3, 6, 9, 12, and 14. The steel frame model reproduces the same behavior under all cases of sudden column removal, as observed with the original structure from Gerasimidis and Sideri [39]. Especially, two critical plastic hinges at the adjacent elements of the column loss and at the top side columns appear with the removal cases at the third (rB-3) and sixth (rB-6) floors. The observed data points, displayed in Fig. 8, have a big range of DCR value, stretching from around 1.00–6.50. Most of the 954

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Fig. 7. A validation 15-floors steel structure from Gerasimidis et al. [40].

Fig. 8. Prediction for the CMRV of a 15-floor steel structure.

scenarios of column removal on the B-axis have a DCR value closing the value of one, which indicates the structure has successfully redistributed the excess load from the column loss. The number of structural elements that redistributes the demand plays a role in the outcome of the DCR as well. Hence, the DCR of the 4-story by 3-bay steel frame structure did not reach a DCR ≤ 2. Within the DCR of interest, i.e. DCR ≥ 3, the case rA-12 has the biggest DCR for all the considered V/H. Although the prediction of CMRV does not indicate which scenario of sudden column removal could have a certain DCR value, it captures the overall response of the structure within an acceptable agreement between the observed and predicted data, which validates the proposed formulation of the CMRV.

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Fig. 9. Interaction of CMRV,

min–DCR–R.

4. CMRV minimum 4.1. Estimation of the CMRV,

min

based on the reduction of the potential for progressive collapse

From a mathematical viewpoint, CMRV, min estimation relies on getting the smallest ratio from Eq. (1), which implies the numerator and the denominator should be equal to their respective minimum and maximum values. As previously discussed, MCEV comes from the V/H and MCE. MCE is related to the design conditions, so it is relatively fixed. V/H can be based on the known V/H of a region by studying its past earthquakes, or the distance of the structure from the closest fault by using a predictive method similar to Fig. 2. Thus, MCEV has also a fixed value. The minimum value of λu for all the scenarios of column removal is R, however, R = 1 represents the capacity limit of the structure to resist collapse under any of the sudden column removal scenarios. Therefore, a structure with R < 1 is vulnerable to a column loss. On the other hand, the recommended value of DCR is the limit value that assures the minimum risk for progressive collapse. In this current work, DCR > 3 indicates a high potential for progressive collapse, while DCR < 3 signifies that the structural elements are capable to bridge across the removed column to assure the structural safety by redistributing the exceeded load. This means that the maximum value of DCR that assures the minimum potential for progressive collapse is DCR = 3. In this regard, the use of R = 1 and DCR = 3 in Eq. (1) gives CMRV, min based on the reduction of the potential of progressive collapse regarding V/H ratio. The specific V/H ratios of 0.50, 0.75, and 1.00 give a CMRV, min equal to 3.50, 2.65, and 2.44, respectively. Based on the values limit of the CMRV, min mentioned above, Fig. 9 shows the contour curves of CMRV, min at constant slices of 0.50 obtained for each considered V/H ratio, by varying DCR and R. The bold lines represent the CMRV, min limit, and the grey ones are the other possible values of CMRV, min depending on DCR and R. Each archetype can be evaluated with these contour curves by using their minimum λu, i.e. R, and the lowest DCR of all the sudden column removal scenarios. The combination of the number of the scheme (S) and the used beam cross-section permits to designate the archetypes. The result shows that five archetypes (S2-330; S2-360; S5330; S5-360; and S6-360) are above the CMRV, min limit for all the considered V/H ratios. The archetype S4-360 satisfies the safety requirement for the CMRV, min with V/H equal to 0.50–0.75, while S3-360 only verifies the one with V/H = 0.50. The others archetypes (on the right side of the CMRV, min limit) do not reach the required value of CMRV, min to ensure minimum seismic structural safety against sudden column removal. In view of Fig. 9, it is clear that the evaluation of the CMRV, min limit based on R = 1, and DCR = 3 is conservative. For example, the archetype S4-360 has R > 1 and DCR < 3 in V/H = 1.00, but its CMRV, min is still below the CMRV, min limit. This can be explained by the increasing influence of the vertical ground motion on the failed structure from V/H equal 0.50–1.00. The decreasing value of CMRV, min for the considered V/H ratios reflects this observation. Moreover, Fig. 8 permits to determine the range of R, indicated by the grey dash lines, for DCR = 3 to obtain the CMRV, min limit. This range diminishes with the increase of V/H. The black dash lines represent the maximum DCR, which are above DCR = 3 for all the considered V/H ratios. Knowing the range of R and the maximum DCR allows a precise design that keeps the structure safe and economic. The CMRV, min evaluation based on the reduction of the potential for progressive collapse provides new insights into the interaction of DCR-R and their respective ranges to mitigate structural collapse. Another way to estimate the CMRV, min is from the structural element behavior viewpoint, which particularly 956

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Fig. 10. Illustration of the beam deformation states under column loss.

concerns the beam element.

4.2. Estimation of CMRV,

min

for different beam deformation states under column loss

The sudden column removal puts the structure under a critical situation that forces the structural elements to their limit to redistribute the exceeding load throughout the structure. This sudden column removal is similar to the sudden application of G on the same structure, particularly when the structure sustains significant deformations as a result. This sudden application of G leads to a dynamic effect, where the ductility demands for all deformation states (elastic, plastic, and catenary) up to the maximum dynamic response must be met to avoid failure. The beam element right above the column loss is the one that picks up and transfers the load to the adjacent elements. That is why a simplified model of the beam is often used to illustrate these deformation states. In this work, the deformation states of the beam element from all the sudden column removal scenarios match the deformation from a beam model with tensile catenary action, as seen in Fig. 10. Although this simplified model is for a simply supported beam with axial end restraints with a midspan plastic hinge, the overall deformation and final states observed for the scenarios with side beam still reflect each deformation states even without the catenary one. Therefore, the advantages of the CMRV, min estimation using the beam deformation states are to measure the change in structural safety under sudden column removal and to calculate its limit related to the maximum response of the structure. Based on the CMRV, min–DCR–R interaction, established in the previous subsection, the scenario of sudden column removal that gives R would represent the archetype in the assessment of the beam deformation states. θ delimits the deformations states by focusing the end rotation of the plastic and the two catenary actions (transient and final). The elastic state can be neglected since it does not imply any damage to the beam element yet. Therefore, the deformation at the maximum structural response coincides with

Fig. 11. Probabilistic distributions of CMRV,

min

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Table 4 CMRV, min evaluation from two different approaches. V/H

0.50 0.75 1.00

Beam actions

DCR = 3

PB

CT

CF

R=1

2.98 2.33 2.18

3.31 2.49 2.29

3.50 2.55 2.30

3.50 2.65 2.44

the deformation at the end of the catenary final. By implementing R and DCR of each beam actions into Eq. (1), three distinct CMRV, min values are obtained for each archetype, as displayed in Fig. 11. To generalize the response of all eighteen archetypes, the mean θ is used to delimit each beam actions: the elastic stage (0.00–2.67%rad); the plastic bending (PB) action (2.67–5.64%rad); the “transient” catenary (CT) action (5.64–14.17%rad); and the “final” catenary (CF) action (14.17–21.00%rad). To estimate the values of CMRV, min at each beam actions, the calculated values of CMRV, min are statistically compiled using a probability distribution, called probability density function (PDF), to calculate the probabilities of occurrence of the CMRV, min. Therefore, the CMRV, min value of the deformation states corresponds to the one with the highest probability density. Table 4 displays the CMRV, min change through the different beam actions, and compares the CMRV, min at collapse from two viewpoints in this section. The increase of the CMRV, min from PB to CF correlates with the decrease of the strength of the beam element. In another word, CMRV, min varies depending on the level of structural damage resulting from the column loss. The values of CMRV, min at CF for the considered V/H ratios are almost identical to the CMRV, min obtained from the acceptance criteria. The value of CMRV, min from V/ H = 0.50 is the same while the ones from the other V/H ratios are a decimal below their counterpart. This difference can be the result of the data limitation (Section 3), which previously stated the need of studying each archetype individually to measure the exact final beam rotation (θf). Although the use of a predetermined θ as a collapse criterion reduces the complexity of the investigation, it affects the accuracy of the observed data by underestimating θ at the collapse of some archetypes. Some cases with the beam cross-section HEB360 have this underestimation. This issue implies that the beam elements of these archetypes have not reached their maximum dynamic response, resulting in the depreciation of the CMRV, min. Despite the data limitations, the overall trend of the CMRV, min for each deformation states depicts the necessary structural strength to resist seismic progressive collapse. Therefore, it is possible to measure the seismic structural safety against a sudden column removal using the rotation of the beam element. This measurement permits to design the steel structures based on R and DCR at the same time while taking into account the influence of the vertical ground motions. 4.3. Effects of the column loss location on the CMRV The length of the failed beam can affect the progressive collapse resistance of the steel structures. Depending on the location of the column loss (side or interior) on the investigated schemes, the trend of the CMRV is compiled based on the length of the failed span. The length of the failed beam for the interior column loss is the total distance between the two intact adjacent columns. Fig. 12 displays the influence of the length of the failed beams on the CMRV for both side (a) and interior (b) column losses. The account of the proposed classification of vertical earthquakes (Fig. 3) in the variation of the structural safety permits to identify the seismic collapse resistance of the structure in case of column loss. The CMRV for the side column loss depicts a descending straight line for each selected V/H ratio, where a shorter failed beam is safer than a longer one. By using the interaction CMRV-DCR (Fig. 6), the DCR value of the failed beam increases when the CMRV of the structure decreases. In another word, the long-failed beam is more likely to fail due to the high DCR value, which increases the potential of progressive collapse. On the other hand, the CMRV for the interior column loss forms a descending curve for each selected V/H ratio, where the curves get closer to each other the shorter the length of the interior failed beam is. The statement that the shorter failed beam is safer than the longer one applies also for the interior column loss. However, the shorter interior failed beam has a higher CMRV than the shorter side one. The steel structure with the interior column loss has several structural elements to pick up the extra-load, especially the two adjacent columns that are connected to the end of the failed beams. The side column loss relies only on

Fig. 12. Influence of the column loss on the CMRV (a. side and b. interior columns). 958

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the failed beam and the upper level of the structure to distribute the extra load through the building. The long interior failed beam depicts a soft slope starting from 12 m outward for each selected V/H ratio. The slow descent of the CMRV for the interior failed beam contrasts with the observed quick slope for the side column loss. Although the long interior failed beam has a small CMRV compared to the side one, the interior failed beam has the advantage to absorb the extra-load by the means of the catenary beam actions. These additional mechanisms do not assure the safety of the structure but they can restrain the spread of the local failure. Based on Fig. 12, steel structures with long span (for example a span of 12 m) will collapse in case of a side column loss. However, the same long span structure will have a CMRV approximately equal to 1.50 with a DCR around 5.00, which will be a better outcome than the case of side column loss but still unsafe considering the current knowledge on progressive collapse and structural safety. 5. Conclusions A quantifiable approach measuring the seismic structural collapse under sudden column removal is proposed using the interaction of the vertical collapse margin ratio (CMRV), the demand-to-capacity ratio (DCR) and the robustness (R). Some observations are obtained as the following: 1) With the evaluation of CMRV using the vertical ground motions, the increase of the value of the vertical-to-horizontal ratios (V/H) diminishes the structural safety, i.e. reduces the likelihood of mitigating progressive collapse. The study of eighteen archetypes with 119 scenarios of sudden column removal concurs with this observation. Not only the values of the CMRV were decreasing from a V/H classification C0 to C3, but they are also increasing with the reduction of the DCR, which is the ability of the structure to bridge across the missing load-bearing element. 2) From the regression analysis, the influence of DCR on CMRV is incomplete without the consideration of the ultimate capacity (λu) of the structure under column removal. The ratio of λu and DCR analogically reflects the expression of the conventional CMR, which is the ratio of the intensity measure (IM) of 50% of the ground motions to the IM of the MCE. In one sense, CMR could be a quantitative relation between the structural influence of a set of ground motions and a selected seism for the design. So, the CMRV formulation combines the influence of the seism through V/H and vertical maximum considered earthquake, and the gravity load through the use of both general (i.e. λu) and local (i.e. DCR) behaviors of the structure under sudden column loss. 3) Based on the assumptions and limitations adopted in this study, the recommended value of DCR for the beam elements can be increased depending on R without affecting CMRV, min. This means that the structural safety using the recommended value of DCR can be achieved within a defined range of R. An improved value of DCR at R = 2 for the V/H ratios 0.50–0.75, and R = 1.5 for V/ H = 1.00 can be used in progressive collapse design to reduce the cost of the structure while preserving its structural strength. 4) The estimation of the CMRV, min using the beam deformation states under sudden column removal shows that the increase of the CMRV, min from the plastic bending to the catenary final correlates with the decrease of the strength of the beam element. The values of CMRV, min at the catenary final are almost identical to the CMRV, min obtained from the acceptance criteria. This difference can be the result of the use of a predetermined beam rotation as a collapse criterion, which reduces the maximum dynamic response of some HEB360 beam cross-sections, resulting to the depreciation of the CMRV, min. 5) Based on the assessment of CMRV using side and interior column losses, the structural collapse resistance has a negative correlation with the length of the side failed beam, i.e. a shorter side beam is more likely to mitigate progressive collapse than a larger one. Although the interior failed beam displays the same negative correlation, its CMRV exponentially tends to 1.50 for a V/H = 1, while the one for the side column loss tends to zero. The interior failed column presents the most resistance when related to the length of beam, however, the values of CMRV and DCR close to 1.50 and 5.00, respectively, are still considered unsafe for the current knowledge on progressive collapse and collapse safety. Acknowledgments This research is financially supported by the National Natural Science Foundation of China (Grant No. 51878123); and the Fundamental Research Funds for the Central Universities (Grant No. DUT19G208). References [1] [2] [3] [4] [5] [6] [7] [8] [9]

UFC, Design of Buildings to Resist Progressive Collapse, Unified Facilities Criteria, Department of Defense, US, 2013. GSA, Alternate Path Analysis and Design Guidelines for Progressive Collapse Resistance, General Services Administration, Washington, DC, 2013. FEMA, Quantification of Building Seismic Performance Factors, Federal Emergency Management Agency, Washington, DC, 2009, p. 421. X. Ou, Z. He, J. Ou, Parametric study on collapse margin ratio of structure, J. Cent. South Univ. 21 (6) (2014) 2477–2486, https://doi.org/10.1007/s11771-0142202-2. Y. Yang, X. Ying, B. Guo, Z. He, Collapse safety reserve of jacket offshore platforms subjected to rare intense earthquakes, Ocean Eng. 131 (2017) 36–47, https:// doi.org/10.1016/j.oceaneng.2016.12.010. L. Xian, Z. He, Y. Zhang, Acceptable structural collapse safety margin ratios based on annual collapse, Eng. Mech. 34 (4) (2017) 88–100. C.B. Haselton, A.B. Liel, G.G. Deierlein, B.S. Dean, J.H. Chou, Seismic collapse safety of reinforced concrete buildings. I: assessment of ductile moment frames, J. Struct. Eng. 137 (4) (2011) 481–491, https://doi.org/10.1061/(ASCE)ST.1943-541X.0000318. L. Xian, Z. He, X. Ou, Incorporation of collapse safety margin into direct earthquake loss estimate, Earthq. Struct. 10 (2) (2016) 429–450, https://doi.org/10. 12989/eas.2016.10.2.429. M.A. Bezabeh, S. Tesfamariam, M. Popovski, K. Goda, S.F. Stiemer, Seismic base shear modification factors for timber-steel hybrid structure: collapse risk assessment approach, J. Struct. Eng. 143 (10) (2017) 04017136, , https://doi.org/10.1061/(ASCE)ST.1943-541X.0001869.

959

Engineering Failure Analysis 105 (2019) 945–960

J.S. Tantely and Z. He

[10] GSA, Progressive Collapse Analysis and Design Guidelines for New Federal Office Buildings and Major Modernization Projects, General Services Administration, Washington, DC, 2003. [11] B.I. Song, H. Sezen, Experimental and analytical progressive collapse assessment of a steel frame building, Eng. Struct. 56 (2013) 664–672, https://doi.org/10. 1016/j.engstruct.2013.05.050. [12] J.M. Weigand, Y. Bao, J.A. Main, Acceptance criteria for the nonlinear alternative load path analysis of steel and reinforced concrete frame structures, Struct. Cong. (2017) 222–232, https://doi.org/10.1061/9780784480397.019. [13] U. Starossek, M. Haberland, Approaches to measures of structural robustness, Struct. Infrastruct. Eng. 7 (7–8) (2011) 625–631, https://doi.org/10.1080/ 15732479.2010.501562. [14] J.W. Baker, M. Schubert, M.H. Faber, On the assessment of robustness, Struct. Saf. 30 (3) (2008) 253–267, https://doi.org/10.1016/j.strusafe.2006.11.004. [15] A. Fascetti, S.K. Kunnath, N. Nisticò, Robustness evaluation of RC frame buildings to progressive collapse, Eng. Struct. 86 (2015) 242–249, https://doi.org/10. 1016/j.engstruct.2015.01.008. [16] Y. Bao, J.A. Main, S.-Y. Noh, Evaluation of structural robustness against column loss: methodology and application to RC frame buildings, J. Struct. Eng. 143 (8) (2017) 04017066, , https://doi.org/10.1061/(ASCE)ST.1943-541X.0001795. [17] Y. Nakamura, Clear identification of fundamental idea of Nakamura's technique and its applications, 12th World Conference on Earthquake Engineering, Auckland, New Zealand, 2000. [18] H. Ghofrani, G.M. Atkinson, Site condition evaluation using horizontal-to-vertical response spectral ratios of earthquakes in the NGA-west 2 and Japanese databases, Soil Dyn. Earthq. Eng. 67 (2014) 30–43, https://doi.org/10.1016/j.soildyn.2014.08.015. [19] N.N. Ambraseys, K.A. Simpson, Prediction of vertical response spectra in Europe, Earthq. Eng. Struct. Dyn. 25 (4) (1996) 401–412, https://doi.org/10.1002/ (SICI)1096-9845(199604)25. [20] A.S. Elnashai, A.J. Papazoglou, Procedure and spectra for analysis of RC structures subjected to strong vertical earthquake loads, J. Earthq. Eng. 1 (1) (1997) 121–155, https://doi.org/10.1080/13632469708962364. [21] Y. Bozorgnia, K.W. Campbell, Ground motion model for the vertical-to-horizontal (V/H) ratios of PGA, PGV, and response spectra, Earthq. Spectra 32 (2) (2016) 951–978, https://doi.org/10.1193/100614eqs151m. [22] E. Akyuz, Development of Site-Specific Vertical Design Spectrum for Turkey, Civil Engineering, Middle East Technical University, Turkey, 2013, p. 83. [23] S. Akkar, M.A. Sandıkkaya, B.Ö. Ay, Compatible ground-motion prediction equations for damping scaling factors and vertical-to-horizontal spectral amplitude ratios for the broader Europe region, Bull. Earthq. Eng. 12 (1) (2014) 517–547, https://doi.org/10.1007/s10518-013-9537-1. [24] J.K. Kim, J.H. Kim, J.H. Lee, H. TaeMin, Development of Korean standard vertical design spectrum based on the domestic and overseas intra-plate earthquake records, J. Earthq. Eng. Soci. Korea 20 (6) (2016) 413–424, https://doi.org/10.5000/EESK.2016.20.6.413. [25] H.M. Liu, X.X. Tao, R. Zhi, X. Wang, Design spectra for vertical earthquake action in the new version of guidelines for seismic design of highway bridges of China, in: N. Korozlu, B. Pokharel, A. Abourriche, D. Salvi, A.I. AlMosaw, S. Chotisuwan (Eds.), International Conference on Innovative Material Science and Technology, Atlantis Press, Paris, 2016, pp. 508–515. [26] M. Khanmohammadi, H. Kharrazi, Residual capacity of mainshock-damaged precast-bonded prestressed segmental bridge deck under vertical earthquake ground motions, J. Bridg. Eng. 23 (5) (2018) 04018016, , https://doi.org/10.1061/(ASCE)BE.1943-5592.0001217. [27] B.I. Song, K.A. Giriunas, H. Sezen, Progressive collapse testing and analysis of a steel frame building, J. Constr. Steel Res. 94 (2014) 76–83, https://doi.org/10. 1016/j.jcsr.2013.11.002. [28] K.P. Campbell, Y. Bozorgnia, Updated near-source ground-motion (attenuation) relations for the horizontal and vertical components of peak ground acceleration and acceleration response spectra, Bull. Seismol. Soc. Am. 93 (1) (2003) 314–331. [29] NIED, High sensitivity seismograph network Japan, National Research Institute for Earth Science and Disaster Resilience, 2003. [30] N.M. Newmark, A Study of Vertical and Horizontal Earthquake Spectra, WASH-1255, Directorate of Licensing, U.S, Atomic Energy Commission, Washington, D.C., 1973, p. 155. [31] K. Kawashima, K. Aizawa, K. Takahashi, Attenuation of peak ground motions and absolute acceleration response spectra of vertical earthquake ground motion, 8th World Conference on Earthquake Engineering, San Francisco, California, 1984, pp. 257–264. [32] N.N. Ambraseys, K.A. Simpson, Prediction of Vertical Response Spectra in Europe, ESEE-95/1, Imperial College, 1995. [33] G. Mohammadioun, B. Mohammadioun, Vertical/horizontal ratio for strong ground motion in the near field and soil non-linearity, 11th World Conference on Earthquake Engineering, Acapulco, Mexico, 1996. [34] A.S. Elnashai, Seismic Design with Vertical Earthquake Motion, Seismic Design Methodologies for the Next Generation of Codes, Balkerna, Rotterdam, 1997, pp. 91–100. [35] J.S. Tantely, Z. He, Seismic collapse margin of steel frame structures with symmetrically placed concentric braces, Intern. J. Steel Struct. 17 (3) (2017) 969–982, https://doi.org/10.1007/s13296-017-9009-6. [36] ASCE-7, Minimum Design Loads for Buildings and Others Structures, American Society of Civil Engineers, Reston, VA, 2010, p. 658. [37] V. Terzic, Modeling SCB Frames Using Beam-Column Elements, Berkeley, CA, (2013). [38] J.S. Tantely, Z. He, Collapse safety margin-based design optimization of steel structures with concentrically braced frames, Adv. Steel Constr. 15 (2) (2019) 137–144, https://doi.org/10.18057/IJASC.2019.15.2.3. [39] S. Gerasimidis, J. Sideri, A new partial-distributed damage method for progressive collapse analysis of steel frames, J. Constr. Steel Res. 119 (2016) 233–245, https://doi.org/10.1016/j.jcsr.2015.12.012. [40] S. Gerasimidis, C.D. Bisbos, C.C. Baniotopoulos, Vertical geometric irregularity assessment of steel frames on robustness and disproportionate collapse, J. Constr. Steel Res. 74 (2012) 76–89, https://doi.org/10.1016/j.jcsr.2012.02.011.

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