DOI: 10.1177/1369433216649385

The collapse probability index

The progressive collapse resistance is the ability of the overall structure to resist local failure. In Equation (1), the probability of an accident P[H] and the conditional probability of the initial local failure P[D/H] are independent of the structural collapse resistance (Starossek and Haberland, 2011). Thus, it can be assumed that P[D/H]P[H]=1 and that only the conditional probability of the progressive collapse P[Collapse/D], namely the collapse probability after the initial failure, is used to evaluate the structural progressive collapse performance.

In a study of earthquake-induced collapse, Zareian and Krawinkler (2007) proposed a collapse probability to evaluate the vulnerability of structures under seismic action. In their study, a total of 40 seismic ground motions were chosen as the standard input, and the collapse probability was defined as the ratio of the number of ground motions that would induce collapse to the total 40 ground motions. When representative ground motions are selected, for example, covering different frequency spectra and durations, the collapse probability of the structures subjected to these seismic actions can then reflect the differences of the seismic resistance of these structures. During progressive collapse analysis and design of building structures (General Services Administration (GSA), 2003; Department of Defense (DoD), 2010), representative vertical support components on different floors are removed to investigate the structural collapse response due to the local failure. Similar to the earthquake-induced collapse probability, when the number of removal scenarios is sufficient and the removed structural members are representative, the progressive collapse resistance of the structural system can then be reflected by the ratio of the number of collapse scenarios to the total number of removal scenarios. Therefore, the progressive collapse probability, i.e., P[Collapse/D], can be expressed as:

where n_c and n are the number of the collapse scenarios and that of all the removal scenarios, respectively. Note that in the analysis, considering the characteristics of loading and structural symmetry, only the representative components must be removed; this in turn significantly reduces the computational workload. For the RC frames with regular structural configuration, the typical columns (e.g., a corner column, two edge column and an interior column) can be selected as the representative components, according to the GSA and DoD guidelines (GSA, 2003; DoD, 2010). For the RC frames with irregular structural configuration, in addition to the typical columns, more columns should also be chosen as the representative components based on a thorough progressive collapse risk analyses, following the corresponding specifications such as the European Committee for Standardization (ECS, 2006). For example, the key columns that are unlikely to be bridged over by the residual structural system should be identified as the representative components through the alternate load path analysis. Further, the columns that are vulnerable to disastrous actions should also be identified through the risk analysis. The collapse criterion regulated by the DoD guideline (2010) is adopted in this study, in which the entire structures are considered to have collapsed when the remaining structures (i.e., the initially non-damaged portion) exceed the allowable structural deformation. For the RC frame structures investigated in this study, the allowable structural deformation refers to the rotation of beams and columns not exceeding 0.2 rad (DoD, 2010).

Progressive collapse fragility curve

Using the method described above (Equation (2)), the structural collapse probability, namely, how likely collapse will be triggered in a structure under the action of the design gravity load, can be obtained. Note that, due to the random nature of the load acting on the structure, the collapse probability under the design gravity load cannot comprehensively assess the progressive collapse resistance of building structures. To address this issue, the redundant structural collapse resistance is further considered in research, engineering design and optimization. In earthquake engineering, the collapse fragility curve is widely used to evaluate the safety margins against earthquake-induced collapse of structural systems (Applied Technology Council (ATC), 2009). The incremental dynamic analysis (IDA) (Vamvatsikos and Cornell, 2002) is used to obtain the fragility curves. In other words, the structural collapse probabilities under different intensities of earthquakes are obtained by gradually increasing the earthquake action to the structure. Consequently, referring to the concept of IDA, the progressive collapse fragility curve of building structures can also be established to evaluate the structural safety margins against progressive collapse (i.e., the redundant resistance). Specifically, during the progressive collapse analysis, by gradually changing the magnitudes of the external load (i.e., the gravity load) acting on the structural system, the corresponding collapse probabilities under different loading levels can then be calculated to establish the progressive collapse fragility curve.

It is should be noted that assessing the collapse probabilities under different loading levels adds significant values not only to scientific research but also to engineering practice. Severe progressive collapse accidents may have been caused by unexpected over-loading. As a typical example, the Sampoong Department Store, which was a 5-story flat plate structure, underwent a progressive collapse in 1995 due to an over-loading condition (Gardner et al., 2002). Such over-loading triggered the punching shear failure at the column-slab joint and the aftermath disproportional collapse of the entire structure, resulting in 502 death, 937 injury and US$216M worth of property damage (Gardner et al., 2002). Given that the gravity load plays a critical role in inducing progressive collapse, it would be more comprehensive to consider the collapse probabilities under different gravity load levels thereby covering a large range of the practical situation.

An example of the progressive collapse fragility curve is illustrated in Figure 1, where the vertical axis is the collapse probability index presented in Equation (2), and the horizontal axis is the selected intensity measure (IM), namely, the nominal gravity. When IM = 1.0G (G is the gravity load), the corresponding collapse probability is the collapse probability subjected to the design load. For a structure with insufficient collapse resistance, the insufficiency can be represented by the part of the curve where IM is less than 1.0G. Similarly, the redundant resistance of the structural system can be represented by the part of the curve where IM is greater than 1.0G. For a structure that is designed to prevent progressive collapse and completely meet the collapse resistance demand, the collapse probability should be zero under any expected initial local failure, (i.e., P_c = 0 when IM = 1.0G), which is shown in Figure 1 as the “Robust structure”. In this case, the entire fragility curve describes the progressive collapse resistance margin of the structure.

Numerical examples of proposed probability index and fragility curve

An 8-story RC frame structure (Model A) is analyzed to illustrate the proposed collapse probability index and the progressive collapse fragility curve. The perspective and plan view of Model A are shown in Figure 2. According to the regulations specified in the GSA (2003) and DoD (2010) guidelines, the representative removed columns (i.e., a corner column, two middle edge columns and an interior column) on each story are selected (see Figure 2b). Each removal column represents a number of the columns at the similar location of the structural plan. The number of these represented columns is considered in Equation (2) when calculating the collapse probability index. The height of the first story is 4.2 m, and that of the remaining stories is 3.6 m. The structure is fully fixed to the ground at the first floor. Tables 1 and 2 list the structural member sizes, material strengths and the loading information. Note that these tables also include additional information for subsequent analyses presented in Section 6. Model A is designed according to the Chinese Code for the Design of Concrete Structures (GB50010-2010) (The Ministry of Housing and Urban-Rural Development of the People’s Republic of China (MOHURD), 2010a) and the Code for Seismic Design of Buildings (GB50011-2010) (MOHURD, 2010b).

Table 1.  Structural parameters of the RC frames

Table 2.  Design loads and seismic action on the RC frames

*PGA (peak ground acceleration) of the design earthquake (i.e., a 10% probability of exceedance in 50 years)

The numerical model of the 8-story RC frame is built using the fiber element model THUFIBER developed at Tsinghua University (Lu et al., 2013), as shown in Figure 3a. In this model, the cross sections are divided into a number of individual fibers which satisfy the plane section assumption. Uniaxial constitutive laws for steel and concrete are assigned to different fibers. This model can accurately simulate the nonlinear behavior of the RC beam or column members under large deformations, for which the dominant failure is flexural or tensile failure. Details of the model including the basic assumptions, elemental failure criteria and modeling techniques can be found in the published literature (Lu et al., 2013). This model has also been validated by the typical progressive collapse tests of RC frame structures (Ren et al., 2014; Sadek et al., 2011; Yi et al., 2008), as shown in Figure 4, where L is the clear span of the tested frames and P_bmax is the peak load of the frames under the beam mechanism. It can be seen that both the beam action and the catenary action developed in the RC frames can be accurately simulated. This model has also been successfully applied in the progressive collapse analyses of building structures (Li et al., 2011, 2014a, 2014b; Ren et al., 2015). In the present study, this element model is used to simulate the beams and columns of the RC frames, as shown in Figure 3b.

An accurate nonlinear dynamic AP (NDAP) analysis is conducted to obtain the structural collapse probabilities subjected to different normalized vertical loads, which are marked as squares in Figure 5 under 32 nominal gravity levels. Eads et al. (2013) proved that structural collapse probabilities under earthquake action follow a lognormal distribution. To better explain the variation of the progressive collapse probability subjected to the nominal gravity, the lognormal distribution function is also used herein to fit the progressive collapse fragility curve, as shown in Figure 5. The fitted curve passes the Kolmogorov–Smirnov test at a 5% significance level, confirming that the structural collapse probability under the progressive collapse condition also follows a lognormal distribution (Benjamin and Cornell, 1970). In view of this, the lognormal distribution is used to fit the progressive collapse fragility curves thereafter.

The simplified assessment method

Following the NDAP analysis, the collapse probability index is calculated to determine whether collapse will occur after the initial failure of local components (GSA, 2003; DoD, 2010). It should be noted that the nonlinear dynamic analysis is very time-consuming. In particular, when establishing the progressive collapse fragility curve, the structural dynamic responses to the initial failure must be calculated at various locations and under different nominal gravity loads, which results in an excessively high computation workload. For the case shown in Figure 5, a total of 1024 nonlinear dynamic analyses are required to obtain the fragility curve. Thus, a simplified assessment method based on the nonlinear static pushdown analysis is proposed in this section to accelerate the calculation process of the collapse fragility curve of RC frames.

In the nonlinear static pushdown analysis, a gradually increased load, i.e., a pushdown load (see Figure 6a) is applied to the collapse-resisting substructure, in which the critical vertical component is removed, to obtain the structural resistance-deformation curve (Khandelwal and El-Tawil, 2008) (Figure 6b). Note that the gravity load acting on the remaining structure remains unchanged. The maximum load on the curve within the allowable structural deformation refers to as the maximum collapse resistance R_max of the substructure after the component fails. For a RC frame structure shown in Figure 6a, the maximum collapse resistance R_max may be achieved under the beam mechanism or the catenary mechanism (Li et al., 2011, 2014a, 2014b), represented by the two types of structural resistance curves as indicated in Figure 6b.

Progressive collapse is induced by the unbalanced gravity load G (which is supported by the initial failed component) acting on the remaining collapse-resisting substructure. Note that most initial damages caused by an accident are instantaneous (e.g., explosion and collision), therefore dynamic effects do exist in the collapse process which amplify the internal forces in the structure. This results in D_min≧G, in which D_min is the minimum demand of the structure to effectively resist progressive collapse. If the maximum resistance R_max is larger than or equal to the minimum resistance demand D_min, then the structure will not collapse. Otherwise, if R_max is less than D_min, the structure will collapse, as is shown in Figure 6b.

The structural minimum resistance demand D_min can be approximately calculated by the following equation:

where DAF is the dynamic amplification factor which is also named as the dynamic increase factor (DIF) in the DoD guideline (2010). Note that DAFs for the demands under the beam mechanism and the catenary mechanism are distinctly different. Hence, the value of DAF should be assessed according to the types of the structural resistance curves (the maximum resistance under the beam mechanism or the catenary mechanism as shown in Figure 6b). GSA (2003), DoD (2010) and Li et al. (2014b) provide the DAF values for RC frame and shear wall structures under the beam mechanism, and Li et al. (2014a) suggested a DAF value for RC frame structures under the catenary mechanism. These proposed DAF values can be utilized to evaluate the structural minimum resistance demand D_min which in turn assess if progressive collapse would occur.

    Note that R_max can be obtained by the nonlinear static pushdown analysis. The consequence of initial local failure can be evaluated by comparing the values of R_max and D_min. Based on the above analysis, the simplified progressive collapse resistance assessment method is proposed:

(1)                Establish a numerical model of the overall structure and conduct a nonlinear static pushdown analysis to obtain the maximum resistance R_max under a representative removal scenario.

(2)                Calculate the DAF value based on the existing formulas (GSA, 2003; DoD, 2010; Li et al., 2014a, 2014b), and substitute the nominal gravity into Equation (3) to calculate the structural minimum resistance demand D_min. Compare R_max and D_min to determine whether progressive collapse will occur under this removal scenario.

(3)                Repeat Step (1) and Step (2) for all representative removal scenarios under the nominal gravity. Based on all the analysis results, calculate the collapse probability index P_c of the overall structure according to Equation (2).

(4)                Continuously increase the nominal gravity and repeat Step (1) to Step (3) to obtain different values of the structural collapse probability index P_c under a series of nominal gravity levels. Plot the progressive collapse fragility curve, as illustrated in Figure 1, and then conduct the progressive collapse resistance assessment.

The proposed method can significantly reduce the computational effort because: (1) the calculation workload of the nonlinear pushdown analysis is significantly less than that of an equivalent nonlinear dynamic analysis; and (2) for each representative removal scenario, only a single nonlinear static pushdown analysis is required to be conducted. In addition, after the value of R_max is obtained, the collapse probabilities under different nominal gravity levels can be evaluated. Thus, the calculation workload is significantly reduced when compared with the NDAP analysis, in which the nonlinear time-history analysis is required for each representative removal scenario under each nominal gravity scenario.

Using the proposed simplified assessment method, Model A is re-analyzed. The value of DAF is essentially important for the accuracy and efficiency of calculating the progressive collapse demand. GSA (2003) and DoD (2010) are two mostly influential design guidelines and engineers are well familiarized with their design procedures. Note that the DAF specified in the GSA guideline is simpler for engineering application due to its independence of the structural ductility as compared to that in the DoD (2010). Therefore, the DAFs proposed in the GSA guideline are selected herein to calculate the structural minimum resistance demand D_min. Also note that the actual DAF in the GSA guideline is combined by a nominal DAF and a demand capacity ratio (DCR) which considers the dynamic effect and the nonlinear material behavior during the collapse process respectively. Two values of DAF, 1.0 and 1.33, are suggested in the GSA guideline. Extensive existing analysis results show that the DAF of frames is between 1.2 and 1.5 (Ruth et al., 2006; JalaliLarijani et al., 2013). Hence, the value of 1.33 is chose as the DAF of the numerical examples in this study. Although such a fixed DAF value may not be the most accurate one, it is quite simple, widely accepted by engineers, and accurate enough that will be proved in the following analysis. 

The lognormal distribution function is also used to fit the progressive collapse fragility curve, as presented in Figure 7. The collapse fragility curve obtained by the simplified method is found to conservatively agree with that obtained by the NDAP method. The average difference of the collapse probabilities is 10.8% between the predictions of the proposed simplified method and those 32 numerical results given in Figure 5. Note that the accuracy of the simplified method is influenced by the DAF value. When the error of the DAF value is acceptable for engineering design, such as those suggested by the DoD and the GSA guidelines, the progressive collapse resistance assessment results are considered acceptable. In addition, if all the structures are evaluated with the same probability index and the same analysis method, the relative value of the assessment results is still convincing. Thus, for highly important structures, the accurate method, i.e., the NDAP method, is recommended to assess the progressive collapse resistance. In the design of normal structures or preliminary analyses of highly important structures, the proposed simplified method is recommended due to its simplicity and convenience.

Numerical examples using the proposed assessment method

Frame model information

Based on Model A, two new RC frame models, named as Model B and Model C, are generated, with detailed information presented in Table 2. The two models are consistent with Model A in structural layout and dimensions, material strength and site soil conditions. However, two different seismic design intensities (meaning that the values of seismic load are different) are considered in the structural design of Models B and C. Subjected to different design seismic loads, i.e. different progressive collapse resistance, the reinforcement ratios are different in Models A, B and C. For each structure, two numerical models are established: Model X–S and Model X, with and without considering the slab effect, respectively; where X represents different seismic action, i.e., A, B and C (see Table 2). This results in a total of six models for the 8-story frame structure. The frame beam is modeled by a T-shaped or a L-shaped fiber beam element where its flange represents the slab (Sasani, 2008). The capacity of the fiber beam model to simulate a T-shaped or a L-shaped RC structural element is validated through comparison with experimental tests of RC columns (Shah, 1984; Sun, 2006), as shown in Figures 8 and 9. Note that the catenary action through the beams and their flanges can be considered in this model whereas the tensile membrane action in the slabs is neglected. By comparison of the analytical results of these models, the performance of the proposed collapse probability index and fragility curve under varying structural design parameters can be evaluated, and thus the rationality and reliability of the proposed index and curve can be validated.

In addition, a 4-story frame structure (Model D) is also designed in this work. Model D and Model A have the same seismic design intensity (i.e., a design PGA = 0.2 g for 10% probability of exceedance in 50 years) and the same overall floor area (approximately 3600 m²). However, their dimensions, including the number of spans, the span lengths and the sectional sizes of members, are different, as is presented in Table 1 and Figure 10. The other design parameters remain unchanged. In view of the structural layout, Model D is symmetric about both X- and Y-axes, given that the same moderate span is chosen in both directions. Model A, a tall frame with rectangular plan view and Model D, a low-rise frame with symmetrical plan view, are considered two design options for a new building. Having the same overall floor area, both of these options meet the functional requirement. The proposed assessment method is used herein to determine which option produces a higher progressive collapse resistance.

Frame models with different seismic design intensities

Using the simplified assessment method, all six of the 8-story models are analyzed; their collapse fragility curves are plotted in Figure 11. With an increase in the design seismic intensity, the structural fragility curve is found to shift rightward, which means that the redundant progressive collapse resistance is improved, i.e., the collapse probability of the structural system subjected to the same initial damage is reduced. This outcome is achieved because the seismic design increases the reinforcement ratio and in turn improves the capacity of the structural components, thereby enhancing the progressive collapse resistance. For the frames without considering the slab contributions (Figure 11a), the design PGA is increased from 0.05 g (Model B) to 0.10 g (Model C), while the collapse probability is only decreased by 8% at IM = 1.0G. Whereas, the design PGA is increased from 0.10 g (Model B) to 0.20 g (Model A), with a decrease of 45% in the collapse probability at IM = 1.0G. This is because when the design seismic intensity is relative low (e.g., Models B and C), the amount of reinforcement is primarily governed by the gravity load. Although the seismic action of Model B doubles that of Model C (Table 2), the reinforcement amount in the beams of Model B (17.88 ton) is only slightly higher than that of Model C (14.80 ton). For Model A under a much higher design seismic intensity, its reinforcement amount in the beams (30.8 ton) is much higher. Thus Model A exhibits a significantly improved progressive collapse resistance. A similar phenomenon can also be observed in the frames considering the slab contributions (Figure 11b). The above comparison indicates that the characteristics of the collapse probability and fragility agree entirely with the theoretical analysis. The proposed probability index and fragility curve can well reflect the progressive collapse performance of the overall structure.

Frame models with or without consideration of slab contributions

Figure 12 shows the progressive collapse fragility curves of Model A and Model C. For each model, the slab effects (Model A-S and Model C-S) are also examined. The collapse fragility curves for Models A-S and C-S are found to significantly shift rightward relative to Models A and C. This outcome suggests that considering the slab contribution results in a much larger progressive collapse resistance. An existing experimental study (Qian and Li, 2012) indicates that the slabs can greatly enhance the progressive collapse resistance of RC frame structures, in which the flexural and axial strengths of the beams can be largely improved by concrete flanges and longitudinal reinforcing bars in the slabs. This comparison on the slab effect again confirms the rationality of the collapse probability index and the fragility curve proposed in this work.

Frame models with different structural layouts

The progressive collapse fragility curves for Models A and D with different structural layouts are presented in Figure 13. The curve of Model D is found to be significantly shifted to the left compared with that of Model A. This implies that the collapse resistance of Model D is much lower than that of Model A. Consequently, the collapse probability of Model D is significantly higher than that of Model A under the same level of nominal gravity. For example, when IM = 1.0G, the collapse probability of Model D, which equals 81%, is 40% higher than that of Model A.

To quantitatively evaluate the progressive collapse resistance of the two structures, the NDAP analysis (GSA, 2003; DoD, 2010) is conducted to improve the progressive collapse resistance capacity of the two models by increasing the reinforcing steel in the beams. For Model A and Model D to achieve the progressive collapse-resistant design target (i.e., the collapse probability equals 0% when IM = 1.0G), the percentage increases in the amount of steel are 2.47% and 11.78%, respectively (see Table 3). It is evident that the design target is much easier to be achieved by Model A than by Model D. Such a difference on the progressive collapse redundancy is also clearly illustrated in the fragility curves (Figure 13).

Table 3. Comparison of steel amount in the beams of Model A and Model D

Note: SA – Steel amount in ton; Ic – Increase as compared to the original structures

Conclusions

A collapse probability index is proposed in this paper. Taking into consideration of all possible initial failure modes, the proposed index is thus independent of the initial local failure and can evaluate the structural progressive collapse resistance from the overall structural point of view. Based on this index, the progressive collapse fragility curve is also proposed in this study. Such a curve can reflect the redundant structural collapse resistance through the variation of the collapse probability with the nominal gravity load. In addition, to reduce the workload of the computation and analysis work, a simplified method to obtain the progressive collapse fragility curve of RC frames is proposed to meet the requirement of routine engineering design practice. Results of the numerical examples suggest that the proposed index and simplified method can provide a good assessment of the structural progressive collapse-resisting performance of RC frames and that the obtained results agree well with the theoretical analysis.

Acknowledgement

The authors are grateful for the financial support received from the National Basic Research Program of China (973 Program) (No. 2012CB719703), the National Natural Science Foundation of China (No. 51578018) and the Australian Research Council through an ARC Discovery Project (DP150100606).

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