NUMERICAL INVESTIGATION OF PROGRESSIVE COLLAPSE RESISTANCE OF RC FRAMES Subject to COLUMN REMOVALS FROM DIFFERENT STORIES Yi Li^{ 1}, Xinzheng Lu^{ 2}*, Hong Guan^{ 3}, Peiqi Ren^{ 2} 1. Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education Beijing University of Technology, Beijing 100124, China 2. Key Laboratory of Civil Engineering Safety and Durability of Ministry of Education Tsinghua University, Beijing 100084, China 3. Griffith School of Engineering, Griffith University Gold Coast Campus, Queensland 4222, Australia Advances in Structural Engineering, 2016, 19(2): 314326. Abstract: This paper presents a nonlinear static pushdown analysis to evaluate the progressive collapseresisting capacity curves of typical RC frames under different deformation. Unlike the previous studies in which only a few typical columns, such as a column on the bottom storey, are removed, this study examines the column removal scenarios for various typical locations from different stories. The primary findings are: (1) the Vierendeel action causes different internal forces in the beams of different stories, which reduces the progressive collapse resistance under the beam mechanism and delays the development of the catenary mechanism. This may result in the beams fail successively from one floor to another in a frame system, which differs from the theoretical assumption that the beams are damaged simultaneously on different floors; (2) seismic designs significantly improve the progressive collapse resistance under the beam mechanism, especially for lower stories. However such an improvement is less significant for the catenary mechanism and little improvement is found for the top regions of the frame structures. Further, a nonlinear dynamic analysis is conducted to validate the predicted resistances of the RC frames in satisfying the requirement of collapse prevention. The design parameters as specified in the existing codes are also discussed. Keywords: reinforced concrete frame; progressive collapse resistance; numerical investigation; nonlinear static pushdown analysis; nonlinear dynamic alternative load path analysis, Vierendeel action. If you need the PDF version of this paper, please email to luxinzheng@sina.com 
1. Introduction Progressive collapse is a disproportional collapse of an entire structure caused by initial local failure of a few structural elements due to accidental events (ASCE, 2005). Progressive collapse of building structures has two significant characteristics: (1) It is a mechanical behavior of the entire structural system, in which the collapse spreads throughout a large part of or the entire structure (Starossek, 2007). In resisting a progressive collapse, on the other hand, the primary contributor is the alternative load paths within the structural system (GSA, 2003; DoD, 2010). (2) It is a mechanical behavior of the structure under large deformations. In conventional laboratory tests and numerical studies, when a structural member reaches a certain amount of deformation after the peak load, the member is considered to have failed, and the residual loading capacity is not considered. For example, in the published literature on fire resistance (ISO, 1999) and seismic resistance (ASCE, 2007; Jiang et al., 2014), when the beam deflection reached 1/50 ~ 1/30 the span length, the beam was considered to have failed. This is however not the case for progressive collapse, where large deformation (deflection approaching 1/5 the span length) characteristics of the beams must be carefully examined (GSA, 2003; DoD, 2010). Under large deformations, the strength of the beams may have degenerated significantly, and the loadcarrying mechanism may have changed (e.g., from a beam mechanism to a catenary mechanism) (Li et al., 2011; 2014a; 2014b). These mechanical behaviors of the beams are very important to the progressive collapse resistance of the structure. Significant progress has been achieved in the past decade with the development of experimental technologies and numerical simulation methods which have promoted fundamental research being conducted on progressive collapse of building structures. For the experimental studies, both static and dynamic methods are used to investigate the collapse behavior of structural members and substructural systems under large deformations. Using the static test method, researchers have studied these progressive collapse behaviors of different types of structures and components such as continuous concrete beams (Su et al., 2009), reinforced concrete (RC) or steel beamcolumn subassemblages (Yap and Li, 2011; Sadek et al., 2011), threestorey, fourbay smallscale RC plane frame structures (Yi et al., 2008), singlestorey smallscale RC flatplate structures (Yi et al., 2011) and twostorey fullscale RC flatplate structures (Kokot et al., 2012). The dynamic tests are commonly conducted using a special device as substitute support components, so that the locally failed structural member can be simulated by instantaneously releasing the special device. Qian and Li (2012a; 2012b) conducted static and dynamic tests to study the collapse behavior of a corner substructure of a concrete frame and comparatively analyzed the dynamic effects of the collapse process. Some researchers took the opportunity of demolishing abandoned buildings to study the dynamic progressive collapse resistance of the entire structures (Sasani et al., 2007, 2011; Sasani and Sagiroglu, 2010; Matthews et al., 2007; Song and Sezen, 2009). As a supplement to the experimental techniques, numerical methods have proven to be convenient and efficient for analyzing the progressive collapse behavior of an entire structural system, either statically or dynamically. Specially, numerical methods are suitable for comprehensive analyses of the various factors that influence the progressive collapse behavior and resistance. Typical studies in this area include: composite slabs using refined finite element analysis approach by Alashker et al. (2010); a concrete structure subjected to the impact of an explosion using the same method by Luccioni et al. (2004); a multistorey steel frame with two different types of brace members studied by Khandelwala et al.(2009), where the joints and the structural components were simulated using the macro model and fiber model, respectively; a 20storey steel frame structure with two different lateral resistant systems investigated by Fu (2009); the effects of the number of stories and bays on the progressive collapse resistance of steel frame structures, evaluated by Kim et al. (2009); steel frames with different strengths and stiffnesses studied by Galal and ElSawy (2010), who concluded that the strength of framed beams has a great impact on the progressive collapse resistance of a steel frame; the progressive collapse mechanisms of steel frames exposed to fire studied by Jiang et al. (2014). Further, Kim et al. (2011) used a randomized method to generate different values for such key parameters as the live load, the elastic modulus and the yield strength of beams, columns and braces. The sensitivity of the progressive collapse resistance of the steel frame to these parameters was analyzed. Similar reliability assessment on the damaged RC frame has been conducted by Huang et al. (2014). Kwasniewski (2010) also developed a detailed 3D model to evaluate the progressive collapse resistance of an 8storey steel frame in the Cardington Fire Test. The structural responses of RC structures under instantaneous and gradual removal of columns were also compared by Rahai et al. (2014). The majority of the numerical analysis on the progressive collapse of overall structural systems adopted the conventional nonlinear dynamic alternate path (NDAP) method to obtain the dynamic responses of the structures (e.g., the timehistory responses of the displacement and the internal force after the initial local failure occurred). In addition to the dynamic responses, the progressive collapse resistance of structural systems is also helpful for understanding the progressive collapse mechanism. However, the structural resistance varies with the structural deformation. To evaluate the progressive collapseresisting capacity curves of structures under different deformation, the vertical load applied to the structures (i.e., the gravity), should be incrementally changed in each NDAP analysis, similar with the incremental dynamic analysis (IDA) in the seismic studies (Vamvatsikos and Cornell, 2002). Obviously, that is very timeconsuming. On the other hand, existing numerical research mainly focuses on the collapse response of entire structures after the initial failure of a few representative structural members, primarily the perimeter columns in the bottom floor (GSA, 2003; DoD, 2010). However, accidents may occur anywhere in the building. Therefore, it is necessary to comprehensively study the progressive collapse resistance of structures under different deformation subsequent to initial damages occurring at all possible locations in a structure. This will provide a valuable reference for engineering practices, yet such research is still lacking presently. In RC frame structures with precast slabs, the frames as the subassemblies of beams and columns are the major structural components resisting progressive collapse. In this paper, a typical 8storey RC frame designed in accordance with low seismic action is firstly studied. A nonlinear static pushdown analysis is performed to investigate the progressive collapse resistance of the RC frame with initial local damage at typical locations of different stories. In addition, the RC frame is redesigned in accordance with high seismic action and the effect of seismic design on the progressive collapse resistance is analyzed by comparing the two RC frames with different seismic design intensities. The resistance of the RC frames is also validated via the nonlinear dynamic method, in which the design parameters as specified in the existing codes are discussed. The outcomes of these analyses can be used as references to further develop the design specifications and methods specific to progressive collapse. 
2. The RC Frame Model The RC frame structure studied herein has eight stories. The height of the first storey is 4.2 m, and that of the remaining stories is 3.6 m. A plan view of the RC frame is shown in Figure 1. The bottom ends of the first storey columns are fixed to the ground. Table 1 lists the sectional sizes and material parameters for the structural elements. The frame is designed in accordance with the Chinese Code for the Design of Concrete Structures (GB500102010) (MOHURD, 2010a) and the Code for Seismic Design of Buildings (GB500112010) (MOHURD, 2010b). The main design parameters are given in Table 2. To study the influence of seismic design on the progressive collapse resistance, reinforcement details are designed according to two different seismic design intensities, while the other design parameters (e.g., plan, sectional size, materials, etc.) remain unchanged. This results in two frame models, namely Model A, designed for a lowseismicintensity region and Model B, designed for a highseismicintensity region.
Figure 1 Plan view of the RC frame Table 1. Parameters of the structural members in the RC frame
Table 2. Design loads and action on the RC frame
*PGA (peak ground acceleration) of the design earthquake (i.e., a 10% probability of exceedance in 50 years) The amount of reinforcement in Model B is about twice that in Model A. The additional reinforcement in the beams of Model B is mainly concentrated at the beam ends where the seismic action is large. However, there is no significant difference in the midspan reinforcement in the beams between the two models because such reinforcement is mainly controlled by the gravity load, which is the same for the two models. Note that the RC frame is designed to have a regular structural arrangement which is popular in engineering practices. As such, the conclusions achieved can provide useful fundamental understandings for future analysis and design tasks for progressive collapse prevention, although some of the conclusions may not be applicable for irregular structures (this should be investigated case by case). 
3. Analysis Method 3.1 Fiber beam element model A fiber beam element model named THUFIBER (Li et al., 2011; Lu et al., 2013) is used to build the numerical model of the frame. THUFIBER takes into account the complex interaction mechanisms of the internal forces in the beam sections and has robust material models covering both unloading and reloading paths. Published literatures show that RC frames exhibiting flexural and axial failures under large deformations can be satisfactorily simulated using THUFIBER with a very efficient computational workload (Li et al., 2011; Lu et al., 2013; Ren et al., 2014). 3.2 Nonlinear pushdown method The nonlinear static pushdown method proposed by Khandelwal and ElTawil (2008) is used in this study to analyze the collapse mechanisms and the progressive collapse resistance of the RC frames. First, the initially damaged columns are removed from the RC frame models. Note that only one column is removed in each analysis. Then, an increasing vertical load q is imposed in the damaged region while keeping the design vertical load g unchanged in other regions (as shown in Figure 2). Using this method, the relationship between the internal forces in the structural elements and the structural deformation during the collapse process, from small to large deformation stages, can be obtained and analyzed. Likewise the relationship between the progressive collapse resistance of the structure, i.e. the vertical load q, and the structural deformation can also be established (see the following discussion). All representative columns on each floor, located at the shortedge, longedge, corner, and interior on the plan layout as shown in Figure 1, are considered individually as the initially damaged columns.
Figure 2 Load pattern for pushdown analysis of the RC frames 3.3 Nonlinear static analysis algorithm based on multiple point constraints A softening process occurs when the RC beams transform from the compressive arch mechanism or the beam mechanism to the catenary mechanism. Problems may be encountered with numerical divergence in the computation for this softening process if the model is loaded by a forcecontrolled algorithm. Therefore, the nonlinear static analysis algorithm proposed by Huang (2009) is used in this study. Based on multiple point constraints, this algorithm can shift the loading mode from a forcecontrolled algorithm to a displacementcontrolled one which is robust to obtain structural responses at the unstable softening stage. This allows for an entire structural resistance curve of the RC frames to be successfully established, as facilitated by this algorithm. 
4. Progressive Collapse Resistance of MultiStorey Frames In this section, the progressive collapse mechanisms and resistance of Model A (designed for a lower seismic design intensity) are evaluated through the analysis of the resistance curves as a result of removing typical columns on different stories. The internal forcedisplacement relationships obtained for multistorey frames will facilitate examination of the collapse resistance influenced by the interaction characteristics of multistorey frames. 4.1 Characteristics of the resistance curve Figure 3 shows the resistance curves of Model A, obtained from the nonlinear static pushdown analysis, for various column removal scenarios. The displacement of the joint on top of the removed column D is chosen as the representative deformation parameter, and the relative resistance (i.e., q/g, where q is the applied vertical load, and g is the design vertical load) is chosen as the resistance parameter. The analysis results obtained after the removal of a column on the x^{th}storey are represented by the legend "xth" in the figure. It can be found that with an increase in the joint displacement in the longedge, shortedge and interior areas, the relative resistance q/g develops significantly at the initial stage and then declines significantly after the first peak. The value of q/g reaches its minimum when D=500~850 mm. However, with further increase of the joint displacement, q/g starts to increase again until the second peak. The first peak resistance is provided by the bending moments at the beam ends (viz., the beam mechanism), and the second is provided by the axial tensile force in the beams (viz., the catenary mechanism). Particularly, pushdown analysis of the corner column removal scenario shows that there is only one peak in the resistance curve, indicating that only the beam mechanism works in this area. This phenomenon coincides with the results discussed by Li et al. (2011). Note in this study that, the first peak resistance is referred to as the beam mechanism instead of the compressive arch mechanism because not all the beams are able to provide compressive arch action under small deformations to resist progressive collapse, whereas all the beams can resist progressive collapse by the bending moments developed at the beam ends.
4.2 Resistance under the beam mechanism Figure 3 also indicates that the progressive collapse resistance of Model A under the beam mechanism has the following two characteristics: (1) For the longedge, shortedge and interior areas, if the removed column is located on the top two stories, the relative resistances are generally higher than those of lower storey column removal scenarios at the corresponding locations. The highest resistance is achieved when the top storey columns are removed. In addition, no significant difference exists in the relative resistance if the removed column is located on the lower six stories. (2) For the corner area, no significant difference in the relative resistance can be observed regardless of the storey the removed column is located on. Again for Model A, designed for a lower seismic design intensity, the gravity load dominates the design, and the amount of reinforcing steel in the beams is mainly governed by the dead and live loads acting on the floor. Due to the same design loads applied to each floor, there is little difference in steel amount for different stories. Therefore, in theory, the progressive collapse resistance should also be similar among different stories. To explain the unusually high progressive collapse resistance of the top storey, the internal forces of beams L1 and L2 (see Figure 1) obtained under different column removal scenarios (i.e., the longedge middle column of the xth storey is removed, where x=6, 7, or 8) are analyzed, as shown in Figure 4 and Figure 5, respectively. In the figures, F j i and M j i represent the axial force and the bending moment, respectively, of beams L1 and L2 on the ith storey after the longedge middle column on the jth storey is removed.
Considering the removal of the 8^{th} storey columns, before the joint displacement D of the edge perimeter beam L1 reaches 500 mm the internal force in this beam ends is a combination of the flexural and axial actions (F 8 8, M 8 8), as shown in Figure 4a. However, when columns on the other stories are removed, the internal forces in the beams on different stories are not identical: those in the beams just above the removed column (e.g., (F 6 6, M 6 6) and (F 7 7, M 7 7)) are close to (F 8 8, M 8 8), whereas (F 6 7, M 6 7), (F 6 8, M 6 8) and (F 7 8, M 7 8) are much smaller, as shown in Figure 4c and Figure 4d. This is due to the existence of the Vierendeel action in beams within the multistories. The axial compressive forces in the beams of different stories form a new moment M_{F} to resist the external loads, as shown in Figure 4b. Thus, the same forces in the beams of the upper stories are much smaller. For a single beam, the existence of the axial compressive force can significantly improve its flexural capacity. However, for a multistorey frame subject to the external loads as a whole, only the beams closer to the lower stories and those having higher axial compressive forces (see Figure 4c) can benefit from improved flexural capacities. Hence, with an increase in the number of stories, such an increased, residual flexural capacity will continuously be redistributed or ¡°diluted¡±, leading to a converged resistance capacity. This explains why, in Figure 3, the removal scenarios on the top two stories result in higher relative resistance, while there is little change in the relative resistance for the removal scenarios on the bottom six stories. Yi et al. (2008) also discovered this phenomenon of uneven internal force development among different stories in the progressive collapse test of a threestorey frame, where the reinforcement strain of the bottom beams was larger than that of the upper beams, which agrees with the above discussion. For beam L2 perpendicular to the edge, the difference in axial forces in the beams of different stories is not obvious (Figure 5c) because there is no compressive arch action. The difference in bending moments in the beams of different stories is also insignificant (Figure 5d). Therefore, the Vierendeel action does not exist in beam L2, and the progressive collapse resistance for different column removal scenarios from different stories is almost identical. The mechanism of beams C1 and C2 in the corner area (see Figure 1) is similar to that of beam L2, in that the relative resistance of the top storey is not higher than those of the bottom stories (Figure 3c).
4.3 Resistance under the catenary mechanism Figure 3 further illustrates that, in the catenary mechanism stage, the relative resistances of the removal scenarios on the upper stories are higher than those on the lower stories, for corresponding locations of longedge, shortedge and interior areas. This phenomenon is similar to that in the beam mechanism stage. It can be found from Figure 4 that the internal forces in the beams of different stories are not evenly developed when considering the interaction of different stories. If the removed column is located on the jth storey, the internal forces in the beam on the (j+1)th storey rapidly convert from compression to tension (700 mm ~ 900 mm). However, such transformation of internal forces is delayed slightly (800 mm ~ 1000 mm) on the (j+2)th storey but significantly (800 mm ~ 1200 mm) on the (j+3)th storey. Therefore, the catenary action of the beams cannot be fully developed at the same time. The lower the storey on which the column is removed, the smaller resistance of the catenary action will be provided at the same deformation, due to the delay of the transformation of internal forces in upper storey beams. 4.4 Structural vulnerability due to uneven internal force development Regarding the theoretical models in the existing codes (GSA, 2003; DoD, 2010), e.g. the catenary model in the tie force method, development of the internal force in structural members is assumed to be even from different stories when resisting progressive collapse. Based on this hypothesis, each floor system is considered to independently carry the collapse load acting on the corresponding floor, and in turn, the whole substructure will successfully resist the total collapse load, as shown in Figure 6. However, as discussed in Section 4.2 and Section 4.3, the internal force development among different stories is uneven in the frame system. More specifically, the forces developed in the lower storey beams are larger than those in the upper stories. Hence, the lower storey beams will be damaged prior to the others. After that, the same mechanism will be applied to the remaining upper storey beams, and in turn, they fail successively from one storey to the other. A similar mechanism also presents in the Vierendeel action of the frame beams. Hence, the combined action of the multistorey floors may reduce the progressive collapse resistance of RC frames.

5. Effect of Seismic Design on Progressive Collapse Resistance 5.1 Resistance under the beam mechanism The effect of seismic design on progressive collapse resistance is examined through Model B, designed for a highseismicintensity region. Figure 7 shows the resistance curves for this model obtained after the removal of the typical columns from each storey. It can be found that in contrast to Model A, for which the seismic design intensity is lower, the relative resistance of Model B in the beam mechanism stage increases significantly at various locations from the top to the bottom stories (Figure 7). This difference is because, for RC frames designed with high seismic design intensity, earthquake action is dominant in the design. The beam reinforcement is governed by the horizontal earthquake action therefore the amount of aseismic reinforcement gradually increases from the top to the bottom stories. This increased amount is mainly located in the beam ends; thereby significantly improving the flexural capacity of the beam. This in turn increases the progressive collapse resistance of the RC frame under the beam mechanism. Figure 8 compares the resistance curves of the model for two different column removal scenarios from the top and the bottom stories. It is evident that a stronger seismic design improves the progressive collapse resistance of the bottom floors more than the top ones because the increase in the amount of seismic reinforcement on the bottom stories is much larger.
5.2 Resistance under the catenary mechanism Continuous reinforcement in the beam creates an axial tension under the catenary mechanism. Thus midspan reinforcement is the key factor influencing the progressive collapse resistance of RC frames under the catenary mechanism. Seismic design primarily increases the amount of bending reinforcement at the beam ends, whereas such an increase at the midspan is far less. For the RC frames presented in this study, the amount of midspan reinforcement for Model B increases slightly on the top storey as compared to Model A. In addition, the increase in the axial strength of the top storey beam is also very limited. Therefore, for the top storey column removal scenario, the progressive collapse resistance of these two models is basically the same under the catenary mechanism (see Figure 8). For the bottom frame beams, on the other hand, the amount of midspan reinforcement increases noticeably, and therefore the resistance clearly increases for Model B under the catenary mechanism (see Figure 8). However, such an increase is still less significant than that under the beam mechanism, as shown in Figure 8. Considering delayed development of the catenary mechanism when multistorey beams work together, the interaction of the above aspects leads to such a change in the progressive collapse resistance due to stronger seismic design being less obvious under the catenary mechanism. This is illustrated in Figure 7 for Model B.

6. Assessment of The Design Parameters in The Existing Codes and Validation of The Collapse Resistance of RC Frames The progressive collapse resistance of RC frames can be evaluated via a nonlinear static pushdown analysis. Given that the collapse process exhibits a strong dynamic effect, a maximum structural resistance that is larger than the design vertical load must be attained in order to prevent progressive collapse from happening. In other words, the maximum relative resistance q_{max}/g must be larger than 1.0. Based on this consideration, the resistance of the RC frames is further validated using the nonlinear dynamic alternative load path method (DoD, 2010; Li et al., 2011). For the two models, Figure 9 presents the maximum relative resistance q_{max}/g versus the ductility ratio m for a total of 32 column removal scenarios. For the portion of the frames undergoing large deformations, m is defined as the ratio of the joint displacement corresponding to the maximum relative resistance to the yield displacement (Pujol and SmithPardo, 2009; Tsai, 2010). The expression q_{max}/g refers to as the peak value of each resistance curve presented in Figures 3 and 7. In Figure 9, the hollow and solid marks demote collapse and noncollapse scenarios, respectively, based on the outcome of the nonlinear dynamic alternative load path analysis. It is evident that a collapse can be prevented when q_{max}/g of the RC frames is larger than 1.236. In addition, the minimum values of q_{max}/g to prevent collapse for the corner, short edge, long edge and internal column removal scenarios are 1.296, 1.327, 1.236 and 1.280, respectively. The factors for different column removal scenarios are very close because the ductility ratios of RC beams designed by the Chinese codes (MOHURD, 2010a; 2010b) are similar (varying from 3 to 4). This can also be confirmed by the existing theoretical study (Li et al., 2014a) that the dynamic amplification factor (i.e. the required values of q_{max}/g, for regular RC frame structures) equals 1.33 when the structural ductility ratio equals 4.0.
Figure 9 Design parameters in the existing codes To facilitate practical designs, the linear static and nonlinear static methods are recommended by the existing codes as the simplified approaches to calculate the progressive collapse resistance. In these methods, the resistance directly obtained from the linear static and nonlinear static analyses, which is equivalent to the applied vertical load g in this study, is further corrected using the dynamic amplification factor (DAF) and the demand capacity ratio (DCR), respectively, to account for the dynamic and nonlinear effects. In the GSA guideline (GSA, 2003), the required equivalent values of q_{max}/g, i.e. DAF/DCR, are 1.0 and 1.33, respectively, for typical and atypical structural configurations (see Figure 9). The analysis results presented in Figure 9 demonstrate that the GSA requirement for atypical structures is adequately met whilst that for typical structures cannot be met which would lead to unsafe designs. This is because the dynamic effect is consistently neglected (q_{max}/g =1). In the DoD2005 guideline (DoD, 2005), a DAF of 2.0, considering the nonlinear effect, is used for the linear and nonlinear static analyses in which the required equivalent value of q_{max}/g is also 2.0. The validation shown in Figure 9 demonstrates that the design parameter is over conservative because the structures satisfying such requirement will exhibit elastic behavior after the column removal. In the DoD2010 guideline (DoD, 2005), different values of DAFs are given for the linear and nonlinear static analyses. An expression of the DAF for nonlinear static analyses is regulated by Eq.(1) which is presented by the solid line in Figure 9.
It can be seen that the GSA2003 requirement is higher than that for typical structures but is still unsafe for the RC frames discussed herein. Model A and Model B will collapse under three and two column removal scenarios, respectively. This is because Eq.(1) is obtained from the numerical analyses of typical planar 3storey and 10storey frames, from which the results cannot accurately describe the nonlinear dynamic effect of all types of frame buildings (Marchand et al., 2009). On the other hand, two DAFs are regulated for linear static analyses in the DoD2010 guideline. The forcecontrolled DAF is 2.0 for fragile structures (i.e. m=1). Although rational, this DAF is only suitable for the elastic response when no structural ductility can be considered. The deformationcontrolled DAF, on the other hand, is much larger than 2.0. This is because it is obtained based on the equal deformation demand in which the same deformation is applied to the RC frames in both linear static and dynamic analyses (Marchand et al., 2009). This is however not the focus of this study in which the nonlinear static method is used and the equal deformation demand is not required. 7. Discussion and Concluding Remarks In this study, the progressive collapse behavior of an RC frame with two different seismic design intensities is analyzed using the nonlinear static pushdown method. The following conclusions are drawn and recommendations for future seismic designs against progressive collapse are given: (1) Vierendeel action leads to uneven internal forces in the beams of different stories. This may result in the beams fail successively from one floor to the other in a frame system, which differs from the theoretical assumption in that the beams are damaged simultaneously on different floors. This action also weakens the compressive arch mechanism of the beams, and therefore reduces the relative collapse resistance of RC frames under the beam mechanism. This is particularly true for frames with more stories. In view of this, considering only the pure flexural strength of the beams under the beam mechanism is a conservative and rational approach for design purposes. Furthermore, the catenary action is the prototype model of the tie force method in existing design codes and is assumed to be able to develop fully on all stories. However, Vierendeel action delays the catenary action on upper stories thereby reducing the progressive collapse resistance. Neglecting this effect will result in insufficient resistance to prevent progressive collapse. Hence, this phenomenon is recommended to be considered in future improvement of the tie force method. (2) Seismic design significantly enhances the progressive collapse resistance of RC frames under the beam mechanism. However, such enhancement is not as significant under the catenary mechanism. This is because the axial tension of the catenary mechanism is provided by continuous reinforcement in the beam. Seismic design consideration significantly increases the reinforcement amount in the beam ends however such an increase at the midspan is small. Hence, it is suggested that the progressive collapse resistance can be improved by extending and connecting a portion of the seismic reinforcement which is proven very effective and economical for RC frames constructed in the seismic areas. In addition, seismic design significantly enhances the progressive collapse resistance of the bottom stories. However, such enhancement is limited for the top stories. Thus, the top stories may become the weakest location in progressive collapse designs. This is a very important and useful finding, because more attention is paid to the bottom stories of a structure in the existing design specifications. It is therefore recommended to specifically examine the progressive collapse resistance of the top stories of RC frames to ensure a safe design of the entire structure. (3) Based on the RC frames presented in this study, the validation demonstrates that the GSA simplified static method for atypical structures provides adequate resistance to prevent progressive collapse; whereas that for typical structures is inadequate due to the absence of dynamic effects which would result in unsafe designs. Furthermore, the nonlinear static analysis specified in DoD2010 also leads to unsafe designs for the RC frames concerned and further investigations in this area are needed. Acknowledgement The authors are grateful for the financial support received from the National Basic Research Program of China (No. 2012CB719703), the National Science Foundation of China (No. 51578018, 51208011) and the Australian Research Council through an ARC Discovery Project (DP150100606). 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Yi, W.J., Kunnath, S.K., Zhang, F.Z. and Xiao, Y. (2011). ¡°Largescale experimental evaluation of building system response to sudden column removal¡±. Proceedings of the 2011 ASCE Structures Congress List of Tables Table 1. Parameters of the structural members in the RC frame Table 2. Design loads and action on the RC frame List of Figures Figure 1. Plan view of the RC frame Figure 2. Load pattern for pushdown analysis of the RC frames Figure 3. Progressive collapse resistance curves for Model A Figure 4. Combined action of beams on different stories (beam L1) Figure 5. Combined action of beams on different stories (beam L2) Figure 6. Structural vulnerability due to uneven internal force development Figure 7. Progressive collapse resistance curves for Model B Figure 8. Comparison between the progressive collapse resistance of Model A and Model B Figure 9. Design parameters in the existing codes * Corresponding author, Email: Luxz@tsinghua.edu.cn, Phone (Fax): +861062795364 