Effects of finite element modeling and analysis techniques on response of steel momentresisting frame in dynamic column removal scenarios
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Abstract
Due to the high cost of the experimental progressive collapse tests, numerical simulation has been widely used by researchers. Finite element method is applied in the majority of numerical progressive collapse studies. In this paper, the influences of finite element modeling and analysis techniques including solution procedure, mesh size, element type, column removal time (CRT), damping, strain rate and outputrelated issues on nonlinear dynamic column removal response of a steel framed structure are evaluated in detail. According to the results, mesh size and column removal time have major influence on the structural response in column removal scenarios, while influences of solution procedure and damping ratio on the maximum response are negligible. Considering the strainrate effects results in lower response and the rate of decline mainly depends on column removal time. Results also show that special emphasis should be laid on the accuracy of saving outputs, because a long interval causes significant change in the estimated response and may lead to misleading conclusions.
Keywords
Progressive collapse Dynamic column removal Nonlinear analysis Mesh dependency Strain rateIntroduction
In structural engineering, progressive collapse is defined as the spread of initial local failure from structural member to member, eventually resulting in the collapse of an entire structure or a large part of it. The underlying characteristic of progressive collapse is that the final state of collapse is disproportionately larger than the initial local failure (Ellingwood et al. 2007). Progressive collapse can be initiated by abnormal loads such as aircraft impact, design error, construction error, fire, explosions, hazardous materials, vehicular collision, and accidental overload (Ellingwood et al. 2007).
Historically, progressive collapse attracted the attention of researchers from the failure of Ronan Point, a 22story residential apartment at London, UK, in 1968. Research in progressive collapse has gained momentum after 11 September 2001. In the aftermath, efforts were made to reform the current building code and publish new guidelines and codes (GSA 2003, 2013; DoD 2005, 2009).
Due to the high cost of the experimental progressive collapse tests, numerical simulation has been widely used by researchers. However, experimental progressive collapse study can be found in the literature, especially on reinforced concrete substructures (Lu et al. 2016; Ren et al. 2016). Moreover, there are few experimental tests on steel framed structures mostly on existing buildings (Song et al. 2014) and connections behavior in collapse scenario (Yang and Tan 2012). These research works are mainly statictype experiments. Exceptions can be found in Song and Sezen (2013) and Song et al. (2014).
Numerical study of progressive collapse is performed either by building and structural analysis software packages such as ETABS (Mohamed 2015; Chiranjeevi and Simon 2016), SAP2000 (Kaewkulchai and Williamson 2006; Sheidaii and Jalili 2015), Perform 3D (Tavakoli and Alashti 2013; Kang and Kim 2014) and OpenSees (Kim and Kim 2009; Kim et al. 2009), or by general purpose finite element packages such as Abaqus (Fu 2013; Tavakoli and Kiakojouri 2014), LSDYNA (Kwasniewski 2010; Agarwal and Varma 2014), ANSYS (Pirmoz 2011; Valipour and Bradford 2012) and ADINA (Pujol and SmithPardo 2009; Zhacng et al. 2010). While it is easier to set up a model in first group (due to good preprocessing ability of these softwares), they are not perfect codes for progressive collapse analysis, for example these softwares could not easily simulate the cracking, damage, strain rate and distributed plasticity and also provide limited outputs compared to the second group. Therefore, for precise progressive collapse analysis, using general purpose finite element packages is preferable.
Due to the poor preprocessing ability of general purpose finite element packages and long analysis time, most of the authors prefer 2D macro models using beam element. A good exception is provided by Kwasniewski (2010), in which a detailed 3D finite element macro model using shell elements is presented. One, two and threedimensional substructure models are developed and compared by Liu (2010) to numerical study of progressive collapse. According to the obtained results, the global response of the onedimensional beam element model is close to that corresponding to the 2D shell or the 3D solid models.
The mesh size has major effects on not only the computational time, but also on the accuracy of the results in nonlinear dynamic finite element collapse analysis (Jiang et al. 2015). Usually, the mesh has been refined around critical regions (e.g., connections and damaged spans) to ensure that the stress and strain gradients in such regions are precisely captured (Alashker et al. 2011). When shell or solid elements are used, fully integrated elements are preferable to ensure that hourglass modes do not contaminate the obtained numerical results (Alashker et al. 2011).
Most of the published numerical progressive collapse analyses are based on alternate path method with sudden column loss at the first floor level. The alternate path is a threatindependent method, which means this method ignores the triggering event and considers structural response after the local failure. In recent years, more researchers have focused on progressive collapse due to certain triggering event such as blast (Fu 2013; Tavakoli and Kiakojouri 2013a), impact (Kaewkulchai and Williamson 2006; Kang and Kim 2014) and fire (Agarwal and Varma 2014; Tavakoli and Kiakojouri 2015). Although some authors propose that threatindependent approaches underestimate the dynamic response, the standard procedure in guidelines and codes focus on the threatindependent approaches (Kiakojouri et al. 2016).
The influence of step size on column removal response is investigated by Gerasimidis and Baniotopoulos (2011). Two algorithms were used for the solution of the column loss: the βNewmark and the Hilbert–Hughes–Taylor. Based on the results, the response of the structures changes when the solution procedure changes. These differences are more critical when the time step size of the methods is high, especially when it is close to the time duration of the column removal. The results also show that the response of the structure is underestimated when the step sizes are not low enough. As the time step size reached values close to zero, two algorithms produced the same results, showing that low time step size values are vital for the reliability of both the algorithms (Gerasimidis and Baniotopoulos 2011).
Generally, progressive collapse analysis is performed in rateindependent analysis, but the influence of strainrate effects should not be neglected when the column is removed instantaneously according to some researches (Tavakoli and Kiakojouri 2013a; Chen et al. 2016). Based on their results, the capacity of structures increase meaningfully when the strainrate effect is considered.
The effect of damping ratio on nonlinear dynamic analysis response and dynamic increase factor (DIF) in nonlinear analysis of structures against column removal are investigated by Mashhadi and Saffari (2016). The results of the analysis reveal that DIF is decreased with increasing damping ratio (Mashhadi and Saffari, 2016). According to Jiang et al. (2017), the influence of damping on progressive collapse of steel frames under a localized fire is negligible in the range of damping ratio from 0 to 10%. On the other hand, the effect of strain rate on the structural performance of steel frames is significant for the cases involving dynamic buckling (Jiang et al. 2017). The study by Li et al. (2018) showed that, for a column instabilityinduced collapse mode, the impact of the damping is larger than the influences of the strain rate on the structural response. However, for the failureinduced collapse, the effects of the strain rate are larger than the damping (Li et al. 2018).
According to GSA guideline, it is preferable to remove the column instantaneously, and the duration for removal must be less than onetenth of the period associated with the structural response mode for the vertical motion of the bays above the removed column (GSA, 2013). Numerical study of the effects of duration of column removal is the subject of some papers. According to results, sudden column removal provides larger structural response (Tavakoli and Kiakojouri 2013b; Chen et al. 2016). Furthermore, some authors suggest that the decision about using either sudden column removal or gradual column removal depends on the type of triggering event (Tavakoli and Kiakojouri 2013b).
Although thousands of papers on numerical study of progressive collapse can be found in the literature, a comprehensive study of the influences of finite element modeling and analysis on the obtained response is rare. In this paper, the effects of finite element modeling parameters and analysis techniques including solution procedure, mesh size, element type, column removal time (CRT), damping, strain rate and output related issues are considered in numerical modeling and the results are discussed in detail with special emphasis on evaluation of column removal point (CRP) displacements. The obtained results provide the rationale for finite element modeling and analysis of steel momentresisting frames in dynamic column removal scenarios.
Primary design, loading and finite element modeling
Primary design
Finite element modeling
In the modeling of steel, the elastic part is defined by Young’s modulus and Poisson’s ratio. The plastic part is defined as the true stress versus logarithmic strain. Abaqus calculates values of yield stress from the current values of strain, approximating the stress–strain behavior of steel with a series of straight lines to simulate the actual behavior. In this study, a bilinear model was used. Therefore, the material behaves as a linear elastic material up to the yield stress of the steel. Then, it goes into the strain hardening until reaching the ultimate stress. Figure 3 shows the bilinear stress–strain curve of steel that is used in this study.
Two different column removal scenarios are considered in the numerical study. In Scenario 1, the corner column in the first story is suddenly removed, while in Scenario 2 the center column in the first story is used for dynamic column removal. These scenarios are shown in Fig. 2.
Unless otherwise specified, all results reported in this paper were obtained using linear beam formulation with the size of 0.25 m, 5% damping, implicit analysis and automatic incrementation.
Verification of the finite element model
Procedure for implicit progressive collapse analysis
Numerical analysis methods may be classified as either explicit or implicit. Implicit method is one in which the calculation of the current value in one step is based on the values calculated in the previous step. This method is unconditionally stable even for large steps. The most important disadvantage of the implicit FEA is that this approach requires the calculation of the inverse of stiffness matrixes, and calculation of an inverse matrix is computationally expensive and timeconsuming, especially for nonlinear problem and complicated models (Wu and Gu 2012; SIMULIA 2014).
Procedure for explicit progressive collapse analysis
In an explicit finite element analysis, instead of solving for displacement, the solution is based on acceleration and, therefore, there is no need for inversion of the stiffness matrixes. The explicit method is conditionally stable. Therefore, small time steps should be employed. This time step basically depends on the order of the smallest value of the ratio of the element length to the speed of the wave in the material. These characteristics make this method practical for the phenomena like blast and impact (Wu and Gu 2012; SIMULIA 2014).
In this study, as shown in Fig. 8, the loads (vertical gravity loads and reactions) increased linearly for 3 s until they reached their maximum amounts and then remained constant for 2 s. The concentrated forces were suddenly removed at 5 s to simulate the dynamic column loss.
Results and discussion
Explicit versus implicit
These calculations were performed on 4 processors Intel Corei7 with 16 GB RAM. The analysis time for the implicit method is two to three times larger than that required for explicit analysis, depending on column removal cases. It should be noticed that the procedures for loading and column removal are different in the two mentioned methods, as shown in Figs. 7 and 8. For explicit analysis, a considerable amount of calculations is needed before column removal, and by manipulating this time the explicit method becomes even more efficient.
Element type and mesh dependency
In linear static finite element analysis using beam elements, the size of the elements does not affect the results of the analysis. On the other hand, for nonlinear dynamic analysis, the solution is highly dependent on the mesh configuration, and the influences of mesh size should be considered in the analysis when nonlinearity is expected. Abaqus offers a wide range of beam elements including linear, quadratic or cubic interpolation. In this paper, B21 (linear interpolation Timoshenko beam element) and B22 (quadratic interpolation Timoshenko beam element) with four different mesh sizes are used for progressive collapse analysis.
Influences of mesh size and element type on the maximum displacement of CRP
Type  Linear (B21)  Quadratic (B22)  

Size (m)  0.1  0.25  0.5  1  0.1  0.25  0.5  1 
Scenario 1^{a}  98  96  91  86  99  98  96  92 
Scenario 2^{a}  60  58  56  54  60  60  59  57 
Column removal time
Moreover, there are examples of triggering events (e.g., fire) where “instantaneous” column removal is not rational for them. Therefore, while abrupt column loss usually provides larger structural response, it is not the most rational simulation for every column loss scenario and special attention should be paid to the nature of the triggering event to prevent overestimation of the structural response.
Strainrate effects
Summary of ratedependency analysis
CRT (s)  Rateindependent  Ratedependent  

0.001  0.01  0.1  0.5  0.001  0.01  0.1  0.5  
Scenario 1^{a}  99  96  60  48  85  84  58  47 
Scenario 2^{a}  63  58  33  30  55  52  33  30 
Damping
Request and saving of outputs
Conclusions

In this study, progressive collapse analysis is performed using both explicit and implicit approaches. A very good agreement between the two methods is observed. The results also show that the time step should be significantly lower than the CRT in both procedures.

Mesh dependency and element type analysis shows that for sufficiently fine mesh, quadratic and linear interpolation beam elements produce the same result. When the quadratic beam element is used, larger element size can be used. Moreover, suggestions are provided for effective mesh size using different elements.

In progressive collapse analysis, unlike seismic analysis, maximum response develops generally after few cycles of vibration and, therefore, influences of damping can be safely ignored. Current results show that even for very large damping ratio, maximum response does not change meaningfully.

Despite guideline recommendation, duration of column removal in a real scenario basically depends on the triggering event. Therefore, while sudden column removal provides larger structural response, it is not a rational assumption in every collapse scenario and special attention should be paid on the nature and characteristic of triggering event to prevent overestimation of the structural response.

Results also show that including the strainrate effects changes the responses meaningfully and decreases displacement of the column removal point. The amount of change depends on the column removal time; by decreasing the time of column removal, the rate effects are increased.

According to results, using long intervals for output time steps can lead to significant errors in the estimated responses. It is recommended that saving the outputs should be performed for every one or two analysis increments. Using longer interval in saving outputs may lead to invalid estimation of structural response.
Further study is still required to investigate the effects of finite element modeling and analysis techniques on progressive collapse response of other steel framed structures, specially braced frames and frames with shear wall. Moreover, extensive study is still needed to gain a further understanding of the relation between threatrelated parameters such as column removal time and strain rate to specific triggering event.
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