Int. J. Reliability and Safety, Vol. 6, Nos. 1/2/3, 2012 15 Robustness assessment for progressive collapse of framed structures using pushdown analysis methods Da-Gang Lu*, Shuang-Shuang Cui, Peng-Yan Song and Zhi-Heng Chen School of Civil Engineering, Harbin Institute of Technology, Harbin, China Email: ludagang@hit.edu.cn Email: cshuangshuang@163.com Email: songpengyan@sina.com Email: czg204@163.com *Corresponding author Abstract: The issue of robustness assessment for progressive collapse of structures has been paid much attention to in the community of civil engineering since the 9/11 event. In this paper, both deterministic indices and reliability-based indices are introduced to quantify the robustness of a structure. The conventional deterministic pushdown analysis is improved to take into account the loading scheme for simulating column loss. To consider stochastic system properties, a random pushdown analysis is proposed by an improved point estimation method based on Nataf transformation. By using the developed pushdown methods, the reserve load carrying capacity of the damaged structure is evaluated, the robustness for resisting progressive collapse of the damaged structure is quantitatively assessed and the key element to be removed which is critical to structural global performance is identified. A code-conforming reinforced concrete frame structure is taken as a case study to demonstrate the applicability of the newly developed methods. Keywords: deterministic indices of robustness; reliability-based indices of robustness; progressive collapse; pushdown analysis; global reliability. Reference to this paper should be made as follows: Lu, D-G., Cui, S-S., Song, P-Y. and Chen, Z-H. (2012) ‘Robustness assessment for progressive collapse of framed structures using pushdown analysis methods’, Int. J. Reliability and Safety, Vol. 6, Nos. 1/2/3, pp.15–37. Biographical notes: Da-Gang Lu is a Professor of Engineering Mechanics in the School of Civil Engineering at Harbin Institute of Technology, China. His research focuses on structural reliability, engineering risk analysis, structural dynamics, earthquake engineering and life-cycle civil engineering. He received his PhD in 1999 from Harbin University of Civil Engineering and Architecture, China. Currently, he serves as members of several international academic organisations, including Joint Committee of Structural Safety (JCSS), International Association for Life-Cycle Civil Engineering (IALCCE), International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE) and International Association on Computational Mechanics (IACM). Shuang-Shuang Cui is a PhD student in the School of Civil Engineering at Harbin Institute of Technology, China. She graduated with a Bachelor of Engineering in Civil Engineering from Shenyang University of Technology, Copyright © 2012 Inderscience Enterprises Ltd. 16 D-G. Lu et al. China, in 2005 and received a Master of Engineering degree in Structural Engineering from Harbin Institute of Technology, China, in 2007. Her research interests are in the field of robustness assessment and seismic safety of structures. Peng-Yan Song is a PhD student in the School of Civil Engineering at Harbin Institute of Technology, China. She graduated with a Bachelor of Science in Theoretical and Applied Mechanics from Harbin Institute of Technology, China, in 2006. Her research interests are in the field of seismic reliability and robustness of structures. Zhi-Heng Chen is a Structural Engineer at Guangxi Hualan Design & Consulting Group. He graduated with a Bachelor of Engineering in Civil Engineering from Guangxi University, China, in 2007 and received a Master of Engineering degree in Engineering Mechanics from Harbin Institute of Technology, China, in 2009. His research interests are in the field of progressive collapse of structures and earthquake engineering. 1 Introduction Progressive collapse and robustness of structures under abnormal loads has been regarded as an important design consideration since the collapse of the Ronan Point apartment building in 1968. Recently, interests in this topic have increased as a result of the terrorist bomb attack on the Murrah Federal Building in Oklahoma City in 1995 and the attack on the World Trade Center in New York in 2001. However, the considered accidental loads are mainly focused on explosions, fires or terrorist attacks, while the problem of progressive collapse and robustness of generic buildings under earthquakes has yet been paid little attention to (Ellingwood and Dusenberry, 2005; Ellingwood, 2006; Kaewkulchai and Williamson, 2006; Talaat, 2007). The current seismic design codes (e.g. Chinese National Standard, 2010; CEN, 2002; among others) just specify some particular requirements and analysis approaches for preventing lateral incremental collapse mechanisms in which the building as a whole moves sideway, the vertical progressive collapse of damaged structures has been gained attentions only recently. Gurley (2008, p.19) confirmed that “records of earthquake damage show that earthquakes can also remove supports, often the corner columns causing two-way cantilever mechanism”, and emphasised that “earthquake engineering does need to include recognition of ‘lost column’ events and to incorporate design against progressive collapse”. The progressive collapse of vast buildings in the great Wenchuan earthquake in 2008 demonstrated once again that it is very important to take into account the ability of resisting progressive collapse and robustness of generic structures under rare earthquakes (Ye et al., 2008). Although the robustness of structures in abnormal events has become a world wide research topic, there has been neither a uniform theory of structural robustness assessment nor a general methodology for quantification of the progressive collapse resistance of real complex structures. In this paper, both deterministic analysis and reliability analysis are conducted to quantitatively assess the robustness and the reserve load-bearing capacity of structures. In deterministic analysis, two redundancy indices are Robustness assessment for progressive collapse of framed structures 17 taken to quantitatively assess the robustness of structures, and a criterion for verifying acceptable robustness is given. To obtain the progressive collapse resistance of damaged structures, a pushdown analysis method is improved to consider the effects of the loading scheme for simulating column loss and the instant of element removing as well as the locations of the removed elements. In reliability analysis, a global limit state equation for load-carrying capacity of a structural system is set up first, in which the margin of safety is the difference between the ultimate downward loading capacity of the structural system and the total vertical loads. A semi-analytical approach which integrates an Improved Point Estimation Method (IPEM), pushdown analysis and moment methods for structural reliability analysis is developed to analyse the reliability of multi-storey framed structures subjected to column failure, the reliability-based index of structural robustness is calculated and the lost element critical to structural system performance is identified. Finally, a reinforced concrete framed structure is taken as a case study to illustrate the effectiveness and applicability of the proposed approach. 2 Definition and indices of structural robustness 2.1 Definition of structural robustness From the viewpoint of structural engineers, robustness is generally recognised as the ability of a structure to avoid disproportionate failure consequences triggered by local damage due to accidental events (Menzies, 2005; Agarwal and England, 2008). This concept may also be simply described as insensitivity or tolerance to local damage from extreme loads or abnormal loads (Faber et al., 2006; Starossek, 2006; Baker et al., 2008). There are four key elements in this definition of structural robustness: accidental events, local damage, nonpropotional failure and failure consequence. First, accidental events, also called abnormal events, include various extreme loading, accidental loads, the failure of material and human error in design or construction. Second, local damage is the damage caused directly by accidental events, the damaged area may be a local region, a substructure or local components, and the damage may be the state that a structural component is out of work due to the accident loads or for lack of strength or ductility caused by low quality construction, or the state that some structural components suddenly rupture due to the deterioration or fatigue damage of materials. Third, nonproportional failure means that the final failure consequence is much more severe than the triggering local failure, in other words, a small damage may cause catastrophic failure consequences. Progressive collapse of a structure belongs to such a nonproportional failure that initiates from local damage and propagates as a chain reaction mechanism into a failure that is disproportionate to the local damage caused by the initiating event (Ellingwood and Dusenberry, 2005; Ellingwood, 2006). Fourth, failure consequence includes the direct economic loss and casualties caused by global system failure or local damage, and indirect economic loss caused by downtime due to structural collapse, secondary disasters or even political influences. A measure of structural robustness should consider the above characteristics and arise by comparing the structural performance of the system in the original state – in which the structure is fully intact – and in a perturbed state – in which a prescribed damage scenario is applied (Biondini et al., 2008). 18 D-G. Lu et al. 2.2 Deterministic indices of structural robustness Note that redundancy in systems is closely related to the concept of robustness, therefore some redundancy indices can be taken as metrics of structural robustness. A general and useful index is the Reserve Strength Ratio (RSR), or reserve redundant factor, defined as the ratio of the load carrying capacity (collapse load) of the intact structure to the design load (Feng and Moses, 1986; Frangopol and Curley, 1987; Pandey and Barai, 1997): RSR = Vu Vd (1) where Vu is the ultimate loading capacity of the intact structure; Vd is the design load. From the standpoint of structural safety theory, RSR is nothing but an overall design safety factor (Ditlevsen and Madsen, 1996); while in the field of earthquake engineering, it normally is called overstrength factor (Uang, 1991; Bertero and Bertero, 1999). The RSR is only related to the capacity of the intact structures, so to consider the damage effects some other indices were proposed, amongst these is the Damaged Strength Ratio (DSR), defined as the ratio between the ultimate load carrying capacity of the structure in damage condition and the design load: DSR = Vr Vd (2) where Vr is the ultimate load carrying capacity of the damaged structure. It can be known that if DSR > 1, the ultimate load carrying capacity of the damaged structure is greater than the design load, so the damaged structure will not collapse when carrying the original design load in this case. The relative reduction of capacity from the intact state to the damaged state may be expressed by the Residual Influence Factor (RIF), or residual strength factor, defined as (Feng and Moses, 1986; Frangopol and Curley, 1987; Pandey and Barai, 1997): RIF = Vr Vu (3) Note that the DSR is the product of the RSR and RIF, i.e., DSR = RIF × RSR, therefore DSR is also called Residual Reserve Strength Ratio (RRSR). There are also some other deterministic measures for structural redundancy (Frangopol and Curley, 1987; Pandey and Barai, 1997), in this paper, however, only DSR and RIF are used to quantitatively assess the robustness of damaged structures. If one of the following two conditions is satisfied: DSR > 1 or RIF > 1/RSR, then the structure is deemed as robust enough to prevent from progressive collapse. 2.3 Reliability-based indices of structural robustness To take into account uncertainties inherent in loads, environments and structural systems, a realistic system performance measure should be based on a probabilistic framework and therefore requires modern structural reliability theory. Some probabilistic indices to measure structural redundancy were proposed based on the relation between damage Robustness assessment for progressive collapse of framed structures 19 probability and system failure probability (Frangopol and Curley, 1987; Fu and Frangopol, 1990). Frangopol and Curley used systems reliability approach to give a probabilistic representation of redundancy in damaged structures: IR = β0 β0 − βd (4) in which IR = redundancy index of a structure, β0 = reliability index of the intact structure, and βd = reliability index of the damaged structure. Fu and Frangopol proposed a Redundancy Index (RI) for the probabilistic measure of system redundancy as follows: RI = Pf 0 − Pfd Pf 0 (5) in which RI = redundancy index of a structure, Pf0 = failure probability of the intact system and Pfd = probability of damage occurrence to the system. Lind (1995) proposed a generic probabilistic measure of system damage tolerance based on the comparison of the failure probability of a damaged to an undamaged structural state as a reciprocal of vulnerability of the system: Td = 1 / V = Pf 0 Pfd (6) in which Td = damage tolerance of a system, V = vulnerability of the system. There also exist some other probabilistic metrics of structural robustness. For example, Baker et al. (2008) proposed a robustness index of an engineered system by comparing the risk associated with direct and indirect consequences. However, in this paper, only the probabilistic redundancy measure [equation (4)] proposed by Frangopol and Curley is taken as the reliability-based index of structural robustness. 3 Pushdown analysis methods for evaluating progressive collapse resistance 3.1 Deterministic pushdown analysis Vertical progressive collapse analysis of a structure is generally performed by the Alternative Load Path (ALP) method, i.e., instantly removing one or several primary load-bearing elements, and then analysing the structure’s remaining capability to absorb the damage (Japanese Society of Steel Construction and Council on Tall Buildings and Urban Habitat 2007). There are four methods for Progressive Collapse Analysis (PCA; Marjanishvili and Agnew, 2006): linear-elastic static, nonlinear static, linear-elastic dynamic and nonlinear dynamic methodologies. The linear static analysis procedure is performed using an amplified (usually by a factor of 2) combination of service loads, applied statically, and response is evaluated by Demand to Capacity Ratios (DCR). But it is limited to relatively simple structures where both nonlinear effects and dynamic response effects can be easily and intuitively predicted. Dynamic analysis procedures (linear or nonlinear, especially nonlinear dynamic), although their accuracy is much higher, are usually avoided due to the complexity of the analysis. The nonlinear static analysis method implies a stepwise increase of amplified (by a factor of 2) vertical loads 20 D-G. Lu et al. until maximum amplified loads are attained or until the structure collapses. This vertical pushover analysis procedure often is called ‘pushdown analysis method’. The advantage of this procedure is its ability to account for nonlinear effects, its usefulness in determining elastic and failure limits of the structure, and its ability of complementing the nonlinear dynamic analysis procedure. Therefore, the nonlinear static procedure is often recommended to be used in conjunction with nonlinear dynamic methodology as a supplemental analysis to determine the first yield and ultimate capacity limits, as well as to verify and validate dynamic analysis results. First yield and ultimate capacity of the structure can be used to determine and validate calculated ductility and rotations. The pushdown analyses of a damaged structure can be accomplished in two different ways based on the loading way: uniform pushdown and bay pushdown. In the uniform pushdown analysis (Figure 1), gravity loads on the damaged structure are increased proportionally until the ultimate limit occurs. The failure may occur outside the damaged bays, and thus it might not be possible to estimate the residual capacity of the damaged bay. In the bay pushdown analysis (Figure 2), however, the gravity load is increased proportionally only in the bays that suffered damage until the ultimate limit is reached in the damaged bays (Dusenberry and Hamburger, 2006; England et al., 2008; Khandelwal and El-Tawil, 2008; Kim and Park, 2008; Kim and Kim, 2009; among others). The residual capacity of the damaged bays can be measured in terms of the gravity overload factor calculated at the instance of first failure in the damaged bays. Schematic of uniform pushdown analysis Load Figure 1 Column removed Step Increasing gravity load proportionally over the entire structure (a) (b) ² Schematic of bay pushdown analysis ² Column removed (a) Load Load Figure 2 Step Increasing gravity load proportionally in the damaged bays Step Nominal gravity load in all other bays (b) (c) In this paper, bay pushdown analysis is used to analyse the effects of loading scheme for simulating column loss and the instance of element removing as well as the locations of the removed elements, while uniform pushdown analysis is used to investigate the effects of the proportionally increased load on the undamaged bays. Robustness assessment for progressive collapse of framed structures 21 In pushdown analysis of a damaged structure, the vertical load is selected according to the provisions of General Services Administration (GSA, 2003). The conventional pushdown analysis does not consider the effects of loading scheme for simulating column loss in the process of applying the vertical load; however, it does have effects on the analysis results in the simulation, there are three reasons: First, the damaged column does not completely lose its capacity instantly, in fact the internal forces of the adjacent components are redistributed during the process of yielding to complete loss of its capacity, so a detailed loading scheme for simulating column loss should be used to realise the process of redistribution of internal forces to reflect the loading situation realistically. Second, once the member (usually a column in the first story) is suddenly removed, the stiffness matrix of the system also needs to be suddenly changed. This may cause difficulty in the analytical modelling process, which can be avoided by the loading scheme for simulating column loss. Third, if a very large load is applied instantaneously to a structure, there will be a serious convergence problem in the analysis program. In particular, it will be magnified when the initial error is accumulated to the stage in which the structure undergoes a strongly inelastic state. Therefore, loading scheme for simulating column loss will have direct effects on the analysed ultimate capacity of the structure in consideration. To consider the effects of loading scheme for simulating column loss, the conventional pushdown analysis procedure generally needs to be revised by applying the unbalanced force to the remaining structure in sub-steps (Kim and Kim, 2009). In order to simulate the phenomenon that one load carrying member is abruptly removed, and consider the loading scheme for simulating column loss, the member should be removed in sub-steps while the gravity load remained unchanged. First, all member forces are herein obtained first from the structural model subjected to the applied load; then the structure is re-modelled without a column with its member forces (P, V and M) applied to the structure as lumped forces to maintain equilibrium position as shown in Figure 3a; then the forces with the same magnitude but opposite in direction to the equivalent forces are applied in sub-steps to cancel the contribution of the removed column. In this way the PCA starts from the moment that the structure is already deformed by the applied load, which reflects the loading situation quite realistically. This procedure is equivalent to releasing the failed elements step by step as shown in Figure 3b. Modelling of sudden removal of a load carrying member …… …… = …… M V + M P P Load (a) Loading scheme of sudden removal of a load carrying member Load Figure 3 Load Resistance Step (b) The unbalanced load on the failed point Step V 22 D-G. Lu et al. The loading scheme of the pushdown analysis which considers the effects of loading scheme for simulating column loss is shown in Figure 4: Step 1: Apply the basic load (DL + 0.25LL) to the structure. Step 2: Remove the failed column. Firstly, all member forces are obtained from the structural model subjected to the applied load; then, the structure is re-modelled without a column with its member forces (P, V and M) applied to the structure as lumped forces to maintain equilibrium position, as shown in Figure 3a; finally, the forces with the same magnitude but opposite in direction to the equivalent forces are applied in sub-steps to cancel the contribution of the removed column. Step 3: Apply the additional load (DL + 0.25LL) to the damaged bays until the completion of the loading or divergence of the program. Since it is not known whether the load 2(DL + 0.25LL) can be applied to the structure or not, a load factor α is introduced to represent the load combination α(DL + 0.25LL) that the structure can bear. The instability of the structure in the analysis procedure (i.e. the divergence of the programme) is defined as the control criteria. Figure 4 Schematic of load applying in pushdown analysis Load α(DL+0.25LL) α =2 p0 α =1 ² A B Step C D (a) Vertical load (b) Step Unbalanced load (c) In static analysis approach specified by Unified Facilities Criteria (UFC, 2005), it is allowed to remove the element first, and then to apply the basic load and the additional load. Therefore in this paper, the instant of element removing is also considered in the pushdown analysis. Considering the fact that there are two different loading ways in the static pushdown analyses, the effect of the proportionally increased load on the undamaged bays is also analysed by uniform pushdown analysis. In addition, the effects of the loading steps are considered in pushdown analysis. 3.2 Random pushdown analysis 3.2.1 Point estimation based on Nataf transformation To consider the statistical correlation of the random variables under the condition of known marginal probability distributions and correlation coefficient matrix, the Nataf transformation (Liu and Der Kiureghian, 1986) is introduced into the Zhao and Ono’s Robustness assessment for progressive collapse of framed structures 23 Point Estimation Method (PEM) instead of the Rosenblatt transformation in the original formulations. The computation of the first few moments of random function Z = g(X) is undertaken in the standard normal space by using the inverse Nataf transformation: μ Z = ∫ g ( x ) f X ( x )dx = ∫ g ⎡⎣TN−1 (u) ⎤⎦ ϕn (u)du { (7a) } σ Z2 = ∫ [ g ( x ) − μZ ] f X ( x )dx = ∫ g ⎡⎣TN−1 (u) ⎤⎦ − μZ ϕn (u)du 2 { 2 α rZ σ Zr = ∫ [ g (x) − μ Z ] f X (x)dx = ∫ g ⎡⎣TN−1 (u) ⎤⎦ − μ Z r } ϕ (u)du (7b) r n r = 3, 4 (7c) where μ Z , σ Z2 , α 3Z and α 4Z = mean, variance, skewness and kurtosis of Z, respectively; f X ( x ) = joint PDF of X; ϕn (u) = the joint PDF of n-dimension standard normal random variables; TN (⋅) = forward Nataf transformation and TN−1 (⋅) = inverse Nataf transformation. 3.2.2 Point estimation for single-variable function For single-variable function, Z = g(X), Nataf transformation reduces to iso-probability transformation, x = FX−1[Φ (u )] , then equation (7) can be approximated by using Gauss– Hermite numerical quadrature in standard normal space: μ Z ≈ ∑ Pj g {FX−1[Φ (u j )]} m (8a) j =1 σ Z2 ≈ ∑ Pj ⎡⎣ g { FX−1[Φ (u j )]} − μ Z ⎤⎦ m 2 (8b) j =1 α rZ σ Zr ≈ ∑ Pj ⎡⎣ g { FX−1[Φ(u j )]} − μ Z ⎤⎦ m r j =1 (8c) r = 3, 4 where uj (j = 1,", m) = estimation points; Pj = corresponding weights and m = the number of estimation points, generally is taken as 5 or 7 (Zhao and Ono, 2000). 3.2.3 Point estimation for multiple-variable function For multiple-variable function g (X ) , it is approximated by a non-product function proposed by Zhao and Ono (2000): n Z ≈ g ′(X ) = ∑ ( Z i − Z μ ) + Z μ i =1 (9) 24 D-G. Lu et al. where Z μ = g ( μ) = g ( μ1 ," , μi ," , μn ) (10) Z i = g ⎡⎣TN−1 (ui ) ⎤⎦ = G (ui ) = G (uμ1 , uμ 2 ," , uμ i −1 , ui , uμi +1 ," , uμ n ) (11) where ui = the vector in which only ui is a random variable, while other variables take the corresponding transformed values of their mean values in standard normal space; uμ j ( j ≠ i ) = the j-th element of the transformed vector uμ that corresponds to the vector μ in standard normal space u. Note that we have introduced Nataf transformation into equation (9), so it is different from that proposed by Zhao and Ono, although their forms are the same. Based on equation (9), the first four moments of multi-variable random function g(X) can be estimated by the following equations: n μ Z ≈ ∑ ( μi − Z μ ) + Z μ (12a) i =1 n σ Z2 ≈ ∑ σ i2 (12b) i =1 n α 3 Z σ Z3 ≈ ∑ α 3iσ i3 (12c) i =1 n n −1 n i =1 i =1 j > i α 4 Z σ Z4 ≈ ∑ α 4iσ i4 + 6∑∑ σ i2σ 2j (12d) where μi , σ i2 , α 3i and α 4i = mean, variance, skewness and kurtosis of Z i , respectively, by using point-estimation of single-variable function. 3.2.4 Random pushdown analysis based on the IPEM To consider the randomness and uncertainties that influence the vertical load-carrying capacity, Vr, of structural system, use of the probability approach should be made to analyse and compute the statistical moments of Vr. However, the vertical load-carrying capacity of a structural system generally is a highly implicit function of the basic random variables, therefore the analytical approximation approaches for uncertainty propagation, such as First Order Second Moment (FOSM) method, is difficult to implement in this case. To lessen the computational burden of Monte Carlo Simulation (MCS), in this paper, the above-mentioned IPEM is combined with deterministic nonlinear finite element analysis, herein the pushdown analysis, to compute the statistical moments of the vertical load-carrying capacity of both intact and damaged structures. For a multiplevariable random function with n basic random variables, the total number of the numerical simulation of the IPEM is m × n, where m is the number of the estimated points, generally is taken as 5 or 7 (Zhao and Ono, 2000). Compared with the huge sampling number of MCS (generally N = 105), obviously the sampling number of IPEM can be reduced dramatically. Robustness assessment for progressive collapse of framed structures 4 25 Robustness assessment for progressive collapse of structures using pushdown analysis methods 4.1 Deterministic assessment of structural robustness The deterministic robustness indices can show us in a quantitative manner whether there are enough alternate paths to safely transfer the loads originally resisted by the failed components. The process of deterministic assessment of structural robustness by using the DSR and RIF is as follows: Step 1: Calculate the design load, Vd, according to the design situations. Step 2: Develop a nonlinear finite element model for the intact structure. Step 3: Conduct the pushdown analysis for the model of the intact structure, and determine the ultimate load-bearing capacity, Vu, from the resulting pushdown curve. Step 4: Assuming a key element (e.g. a column) is removed instantly from the intact structure, conduct the pushdown analysis for the damaged structure, and determine the ultimate load-bearing capacity, Vr, of the column-removed structure from the newly resulted pushdown curve. Step 5: Calculate DSR and RIF based on the values of Vd, Vu and Vr. Step 6: Evaluate the robustness of the structure by use of DSR and RIF. Step 7: Change the location of the removed element, repeat the above steps and identify which removed member is critical to the system performance. 4.2 Reliability-based assessment of structural robustness In this paper, the failure probability of multi-storey framed structures subjected to column removal is investigated using a simplified global reliability method (Lu et al., 2008; Song et al., 2009). A global limit state function for vertical load-carrying capacity of a structural system is first set up as the difference between the ultimate downward loading capacity of the structural system and the applied total vertical loads as the margin of safety: Z u = g (Vu , DL, LL) = Vu − DL − LL (13) Z r = g (Vr , DL, LL) = Vr − DL − LL (14) in which Vu = the random ultimate downward loading capacity of the intact structure, Vr = the random ultimate downward loading capacity of the damaged structure, DL = the dead load and LL = the total live load applied to the structure. The formulation of equation (13) or equation (14) is the same as that of a component reliability problem, since the ultimate downward loading capacity of a structure Vu or Vr can be taken as the component resistance, while DL and LL are equivalent to the load effects of the component; therefore the system reliability problem of the damaged structure can be easily solved by First Order Reliability Method (FORM) of a component. However, the statistical information and the distribution formulations of Vu 26 D-G. Lu et al. and Vr are difficult to directly obtain in that they are implicit and nonlinear functions of the random properties. To overcome this difficulty, the above developed random pushdown analysis approach based on the IPEM is used to calculate the first four statistical moments of Vu and Vr. The lognormal distributions of Vu and Vr are assumed herein and verified by MCS. DL is modelled by normal distribution, while LL satisfies Type I extreme value distribution of the maximum variable. To consider higher statistical moments of Vu and Vr, the moment methods (both third-moment method and fourthmoment methods) developed by Zhao and Ono (2001) are applied to solve the failure probabilities of global limit state functions [equations (13) and (14)]. Then the results are compared with those by virtue of FORM and verified by MCS. By combining the IPEM, pushdown analysis and moment methods, the reliabilitybased index of structural robustness is evaluated through this hybrid semi-analytical method with the simplified global limit state functions [equations (13) and (14)] as follows: Step 1: Develop a nonlinear finite element model for the intact structure. Step 2: Determine the probability distribution types and the distribution parameters of random structural properties as well as DL and LL. Step 3: Compute the first four statistical moments of the vertical load-carrying capacity of structures Vu and Vr by the random pushdown method. Step 4: Compute the first four statistical moments of the safety margins Zu and Zr by the IPEM. Step 5: Compute the reliability index of the intact structure, β0, and the reliability index of the damaged structure, βd, by moment methods with respect to equations (13) and (14), respectively. Step 6: Compute the reliability-based index of structural robustness, IR, according to equation (4). Step 7: Change the location of the removed element, repeat the above steps and identify which removed member is critical to the system performance. 5 Case study: a RC frame structure 5.1 Design and modelling of the case study structure To investigate the robustness for progressive collapse of the codified designed structures, a typical three-bay and five-storey reinforced concrete moment frame structure is designed according to the current seismic design code of buildings (Chinese National Standard, 2010). The plan and elevation of the case study structure and the details of structural members are shown in Figure 5. The vertical loads are listed in Table 1. The mechanical properties of the materials used in the structure are: compressive strength of concrete, fc′ = 29.76 MPa; yield strength of reinforcement, fy = 388.0 MPa and the Young’s modulus of concrete and steel, Ec = 3.356 × 104 MPa and Es = 200 GPa, respectively. The fundamental period of the structure is 0.74 s before the gravity load is applied and 0.95 s after the gravity load is applied. Robustness assessment for progressive collapse of framed structures 27 Vertical loads Table 1 Uniformly distributed loads (kN/m) Concentrated loads (kN) Locations Dead load Roof floor Live load External nodes Dead load Internal nodes Live load Dead load Live load 12.09 1.1 121.0 7.0 149.6 10.6 Middle floor 9.35 4.4 102.3 28.1 122.9 42.5 Ground floor 9.35 4.4 105.4 28.1 126.0 42.5 C A5 CD5 A4 CD4 A3 CD3 A2 CD2 6000 B 3600 3600 1 2 3600 3 3600 3600 4 5 A 3600 6 A1 Notes: CD1 6000 6000 2400 D1 7 A (a) AB1 BC1 B1 C1 3300 3300 2400 6000 D 3300 3300 Plan and elevation of the RC frame and details of structural members 3900 Figure 5 B C D (b) (c) C1 represents a column in the first story in column line C; AB2 represents a beam on the second floor in bay AB. The OpenSees (McKenna et al., 2008) is chosen as the simulation platform. Concrete is modelled by Concrete01, confinement is specified implicitly by using the confined stress–strain relationships proposed by Mander et al. (1988); reinforcing steel is modelled by Steel02 with 1% strain hardening. Beams and columns are modelled by BeamWithHinges elements with co-rotational geometric transformation, the length of plastic hinge is defined as the height of the section of the beam. In the pushdown analysis for RC-framed structures, the yielding of the rebar in tension means that the cross-section undergoes the initial plastic stage, corresponding to the yielding state of the structure; the cross section rotation is defined as peak (ultimate) rotation when the strain of core concrete at the edge reaches its ultimate strain, whose corresponding state is defined as the ultimate state; the state of instability of the structure or the divergence of the program is defined as collapse. 5.2 Progressive collapse resistance evaluation by pushdown analysis approach 5.2.1 The effects of the loading scheme for simulating column loss The conventional pushdown analysis procedure is revised to consider the effects of the loading scheme for simulating column loss by applying the unbalanced force to the remaining structure in sub-steps, since it reflects the loading situation quite realistically. 28 D-G. Lu et al. It is observed that the failure of the structure is represented by the occurrence of the plastic hinges at the ends of the beams, and the collapse of the structure is a consequence of the instability of the system. The analysis results indicate that, the initial yield rotations of different elements are not the same, for instance, it is about 0.0027−0.0032 rad at the end of beam BC, while 0.032−0.049 rad at the end of beam AB. However, there is no obvious difference in the ultimate rotations at the ends of different elements, although the ultimate rotations at both ends of beam AB1 are larger than others. The maximum rotation in the analysis is related to the span of the beam, the section-rotation of beam BC is the largest in the process of analysis, it is greater than 0.17 rad, although the maximum section-rotation of beam AB remains at 0.07 rad. Plastic hinge occurs only in one column (A1) in the whole process of the analysis, and the maximum rotation of the column remains at a low level. It can also be seen that, although θu/θy at the right end of beam AB is different from the others, the initial yield rotation of each element is small, the ultimate plastic rotations θu−θy of each element are close to each other, the mean value of θu−θy is 0.0345, the standard deviation is 0.0019 and the coefficient of variation is 0.055. The mean value of ultimate plastic rotation of each element is larger than that in the acceptance criteria recommended in FEMA356 (FEMA, 2000), i.e. 0.0345 rad > 0.025 rad. However, the deformation limit of the rotation degree of components obtained in this paper is close to the limit value recommended in the UFC (0.105 rad), but it is larger than that offered by FEMA356 (0.05 rad). As shown in Figures 6 and 7, the loading scheme for simulating column loss has no effects on the instant of the occurrence of structural yielding rotation angle, ultimate rotation angle and the rotation angle against collapse for the first time (it is corresponding to the load factor in pushdown analysis), the two pushdown curves almost coincide when the load factor is less than 1.69 in Figure 6. If the effects of loading scheme for simulating column loss are not considered (conventional pushdown analysis method), the structure will collapse instantly (it is corresponding to the load factor of 1.72) once there is a rotation angle of members which exceeds the limit value against collapse (it is corresponding to the load factor of 1.69). Otherwise, a larger ultimate load-bearing ability of the structure can be obtained, the structure will collapse when the load factor is 1.92, although the rotation of beam BC is far more than the limit value against collapse at the moment. Therefore, a larger structural robustness will be gained when the effects of loading scheme for simulating column loss are considered, although it has no effects on the failure modes. 5.2.2 The effects of the instant of element removing In the static analysis approach specified by UFC (2005), it is allowed to remove the element first and then apply the basic loads and the additional loads. Therefore in this paper, two schemes of the pushdown analysis are considered. In the first case, the element is removed before the gravity load is applied to the structure in pushdown analysis; while in the second case, the element is removed after the gravity load is applied to the structure, which corresponds to the physical problem considering the initial deformation in the analysis, with the results shown in Figures 7b and 7c. The distributions of plastic hinges of the damaged structure are shown in Figure 8, while the pushdown curves are given in Figure 9. Robustness assessment for progressive collapse of framed structures Figure 6 29 Pushdown curves of the case study structure considering the effects of the loading scheme for simulating column loss 2 Step-by-step failure Instant failure Load factor (α) 1.8 1.6 1.4 1.2 1 0 100 200 300 400 500 Vertical displacement (mm) Figure 7 The effects of the loading scheme for simulating column loss on the failure modes of the damaged structures 30 D-G. Lu et al. Figure 8 Failure modes of the damaged structures considering the effects of the instant of element removing Figure 9 Pushdown curves of the damaged structure considering the effects of the instant of element removing Load factor (α) 1.8 1.6 1.4 Column is removed after gravity load is applied Column is removed before gravity load is applied 1.2 1 0 50 100 150 200 250 300 350 Vertical displacement (mm) As shown in Figures 8 and 9, a larger ultimate ability of resisting progressive collapse of the structure is obtained if an element is removed before the gravity load is applied to the structure in pushdown analysis, although it has no effects on the failure modes. The structure will collapse due to the large plastic rotation angles at the ends of beams in both schemes of pushdown analysis. 5.2.3 The effects of the proportionally increased load on the undamaged bays The uniform pushdown is used to investigate the effects of the proportionally increased load on the undamaged bays. The pushdown curves are given in Figure 10 and the distributions of plastic hinges of the damaged structure are shown in Figure 11. Compared with bay pushdown analysis results (computed by conventional pushdown analysis method) shown in Figures 7b and 7c, we see that it has no effects on the failure modes of the structure by considering the effects of the proportionally increased load on Robustness assessment for progressive collapse of framed structures 31 the undamaged bays, but a larger ultimate load factor is obtained in the analysis. This is because that the internal force values of beams in Bay CD are reduced, due to the redistribution of the internal force caused by the amplified static load which acts on the adjacent undamaged bays. Figure 10 Pushdown curves of the damaged structures considering the effects of the proportionally increased load on the undamaged bays 2 Load factor (α) 1.8 Uniform pushdown Bay pushdown 1.6 1.4 1.2 1 0 100 200 300 400 Vertical displacement (mm) Figure 11 Failure modes of the damaged structures considering the effects of the proportionally increased load on the undamaged bays 5.2.4 The effects of load steps In pushdown analysis of a damaged structure, the vertical loads can be applied to the structure in different load steps. The effects of the load steps are also considered here, the results are shown in Table 2. 32 Table 2 Size of load step D-G. Lu et al. Ultimate load factors obtained by pushdown analysis in different load steps Considering the effects of the loading scheme for simulating column loss Without considering the effects of the loading scheme for simulating column loss 0.001 1.84 1.78 0.005 1.98 1.85 0.01 1.92 1.72 0.02 0.84 2.00 0.05 0.80 0.80 It is concluded that the analysis results obtained by considering the loading scheme for simulating column loss are more stable than that not considering it. The failure modes of different analysis steps are the same, but the ultimate load factors are much different from each other, a smaller ultimate load factor is obtained when the load step is too large or too small. However, different cases demonstrate complete consistency until there is a cross-section rotation which is beyond the limit value against collapse occurred firstly, the deformation limit of the rotation degrees of components for reinforced concrete obtained in this analysis is consistent with that given in UFC. The limit of rotation degree of components is suggested as the major control criteria for the failure of components or for preventing the structure from collapse. 5.3 Deterministic assessment of structural robustness The pushdown analysis is performed on the damaged structure which assumes that the columns in column line B on each floor are removed one by one. The robustness of the damaged structure is obtained using the above described methods. Take the frame with column B1 removed as an example. The nominal load, 1.3(DL + LL), considering the dynamic effects of the removed column is taken in this paper, so the corresponding load factor is 1.30. The load factors for the intact structure and the damaged structure are 17.02 and 1.72, respectively. The deterministic indices of structural robustness are then calculated as follows: RSR = Vu 17.02 = = 13.09 Vd 1.30 DSR = Vr 1.72 = = 1.32 Vd 1.30 RIF = Vr 1.72 = = 0.10 Vu 17.02 The deterministic robustness indices of the damaged structure for column B1, B2, B3, B4 and B5 removal scenarios are shown in Figure 12. It can be seen that for column B2 removal case, the impact on the overall performance of the structure is the smallest; while for column B3 removal case, it is the largest. The smallest value is taken herein as the robustness index of the whole system. Robustness assessment for progressive collapse of framed structures 33 The critical value 1.55 1.5 1.32 1.31 1.33 1.19 1.00 1 0.5 0 B1 B2 B3 B4 Index of robustness ( RIF ) Index of robustness ( DSR ) Figure 12 Deterministic robustness indices of the damaged structures assuming the columns in column line B removed for each floor respectively 0.10 0.10 0.09 0.1 0.10 0.08 0.05 0 B5 The critical value 0.12 B1 B2 B3 B4 B5 The floor in which the column in column line B is removed The floor in which the column in column line B is removed (a) (b) According to the deterministic assessment criteria: DSR = 1.32 > 1 or RIF = 0.1 > 1/ RSR = 1/13.09 = 0.08, this codified structure can be regarded as robust enough to prevent from progressive collapse. However, the global robustness of the structure needs to be further studied due to the different probability of occurrence of different cases. 5.4 Reliability-based assessment of structural robustness The random variables considered in structural properties include: yielding strength fc of concrete; yielding strength fy, elasticity modulus E, and the second stiffness factor α of steel. The probability models and distribution parameters of random structural properties are listed in Table 3 (Song et al., 2009), while the probability models and statistics of vertical loads are listed in Table 4 (Li et al., 2006). Statistics of random structural properties Table 3 RVs Mean value CoV Types fc (N/mm2) 29.8 0.2 Log-normal fy (N/mm2) 388.0 0.08 Log-normal 2 × 10 0.2 Log-normal 0.01 0.2 Normal 2 E (N/mm ) α Table 4 Loads 5 Correlation coefficient 0.3 Statistics of vertical loads CoV Type Dead load Mean value (kN) 3067.86 0.1 Normal Live load 211.82 0.25 Type I largest The random pushdown analysis by using the IPEM based on Nataf transformation is applied to analyse the probabilistic ultimate downward loading capacity of the intact RC frame and column-removed structure. The first four moments of the total vertical loadcarrying capacity of the intact and the damaged structural systems are computed by the random pushdown method, which are listed in Table 5. 34 Table 5 D-G. Lu et al. The first four moments of the vertical load-carrying capacity of structural systems Structural systems Mean value (kN) Std (kN) Intact structure 16615.67 2457.27 Column B1 – removed structure 4035.70 Column B2 – removed structure 3720.56 Column B3 – removed structure Column B4 – removed structure Column B5 – removed structure Skewness coefficient Kurtosis coefficient –0.19 2.88 490.11 0.14 2.95 398.52 –0.17 2.58 4694.42 482.18 0.24 3.25 4597.01 466.02 0.75 3.86 4069.46 341.93 0.30 2.80 The reliability index of the intact structure calculated by the fourth-moment method is β0 = 7.27. The reliability indices of the damaged structure, assuming that the columns in column line B on each floor are removed one by one by the same method, are listed in Table 6. The results are compared with FORM and verified by MCS, and the reliability indices calculated by the two methods are also shown in Table 6. It is found that the proposed method in this paper has the same accuracy as MCS and FORM, but with great computational efficiency. Table 6 Reliability indices of column-removed structures Columns removed Reliability methods The proposed method B1 B2 B3 B4 B5 1.33 0.87 2.62 2.09 1.74 FORM 1.33 0.87 2.63 2.09 1.75 MCS 1.34 0.85 2.59 2.12 1.71 The reliability-based robustness indices of the damaged structure are shown in Figure 13. It can be seen that for the case of column B2 removal, the impact on the overall performance of the structure is the smallest; while for the case column B3 removal, it is the largest. This result is consistent with that of the deterministic assessment. Figure 13 Reliability-based robustness indices of the damaged structures assuming the columns in column line B removed for each floor respectively Index of robustness ( IR ) 1.56 1.40 1.5 1.22 1.31 1.14 1 0.5 0 B1 B2 B3 B4 The floor in which the column in column line B is removed B5 Robustness assessment for progressive collapse of framed structures 6 35 Conclusions In this paper, the quantitative assessment of the robustness for resisting progressive collapse of framed structures is presented using both deterministic pushdown analysis and random pushdown analysis. The conventional deterministic pushdown analysis approach is improved to take into account the loading scheme for simulating column loss and the instant of element removal; a new random pushdown analysis approach taking into account random system properties is developed by an IPEM based on Nataf transformation. Both deterministic indices and reliability-based indices for structural robustness are reviewed. The deterministic pushdown analysis considers the effects of the loading scheme for simulating column loss and the instant of element removal, different load steps as well as the effects of the proportionally increased load on the undamaged bays, it is found that these factors all have no effects on the failure modes of progressive collapse, but they may have influences on the ultimate ability of resisting progressive collapse of the structure. The reserve load carrying capacity of the damaged structure is evaluated by the improved pushdown method to quantitatively assess the robustness for progressive collapse and to identify which removed member is critical to the system performance. The random pushdown analysis can effectively consider the effects of random structural properties and modelling uncertainty by the IPEM instead of MCS. The reliability-based index of structural robustness is quantitatively assessed by a simplified global reliability approach using a hybrid semi-analytical method combining random pushdown analysis with moment methods. It is found that the new method has the same accuracy as MCS and FORM, but with great computational efficiency. It is further found that the assessment results and conclusions of deterministic-based assessment and reliability-based assessment for structural assessment are consistent with each other. Acknowledgements The financial supports of the first author by the Natural Science Foundation of China through projects (Grant No. 50978080, 90715021 and 50678057) are gratefully acknowledged. We thank the anonymous reviewers for their constructive and critical comments and suggestions. References Agarwal, J. and England, J. (2008) ‘Recent developments in robustness and relation with risk’, Structures & Buildings, Vol. 161, No. 4, pp.183–188. 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